10 46

10_45

10_47

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 46 at Knotilus!
	10_46 is also known as the pretzel knot P(5,3,2).

Knot presentations

Planar diagram presentation	X6271 X8493 X2837 X16,10,17,9 X14,5,15,6 X4,15,5,16 X18,12,19,11 X20,14,1,13 X10,18,11,17 X12,20,13,19
Gauss code	1, -3, 2, -6, 5, -1, 3, -2, 4, -9, 7, -10, 8, -5, 6, -4, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 8 14 2 16 18 20 4 10 12
Conway Notation	[5,3,2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{8, 13}, {1, 12}, {13, 11}, {12, 6}, {10, 5}, {11, 9}, {7, 10}, {6, 4}, {5, 3}, {4, 2}, {3, 8}, {2, 7}, {9, 1}]

[edit Notes on presentations of 10 46]

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 46"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X6271 X8493 X2837 X16,10,17,9 X14,5,15,6 X4,15,5,16 X18,12,19,11 X20,14,1,13 X10,18,11,17 X12,20,13,19

In[5]:=

GaussCode[K]

Out[5]=

1, -3, 2, -6, 5, -1, 3, -2, 4, -9, 7, -10, 8, -5, 6, -4, 9, -7, 10, -8

In[6]:=

DTCode[K]

Out[6]=

6 8 14 2 16 18 20 4 10 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[5,3,2]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(3,\{1,1,1,1,1,-2,1,1,1,-2\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{8, 13}, {1, 12}, {13, 11}, {12, 6}, {10, 5}, {11, 9}, {7, 10}, {6, 4}, {5, 3}, {4, 2}, {3, 8}, {2, 7}, {9, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [3][-15] Hyperbolic Volume 7.717 A-Polynomial See Data:10 46/A-polynomial

[edit Notes for 10 46's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for 10 46's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-6z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, 6 }
Jones polynomial	$q^{11}-2q^{10}+3q^{9}-4q^{8}+4q^{7}-5q^{6}+4q^{5}-3q^{4}+3q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+5z^{4}a^{-8}+11z^{2}a^{-4}-18z^{2}a^{-6}+7z^{2}a^{-8}+6a^{-4}-8a^{-6}+3a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-5z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-4}-23z^{6}a^{-6}-12z^{6}a^{-8}+4z^{6}a^{-10}+5z^{5}a^{-5}-12z^{5}a^{-7}-13z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-4}+42z^{4}a^{-6}+13z^{4}a^{-8}-9z^{4}a^{-10}+3z^{4}a^{-12}+5z^{3}a^{-5}+23z^{3}a^{-7}+9z^{3}a^{-9}-7z^{3}a^{-11}+2z^{3}a^{-13}-17z^{2}a^{-4}-29z^{2}a^{-6}-7z^{2}a^{-8}+2z^{2}a^{-10}-2z^{2}a^{-12}+z^{2}a^{-14}-6za^{-5}-10za^{-7}-2za^{-9}+2za^{-11}+6a^{-4}+8a^{-6}+3a^{-8}$
The A2 invariant	$q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-12}+q^{-14}-2q^{-16}-q^{-18}-3q^{-20}-q^{-22}+q^{-28}+q^{-32}$
The G2 invariant	$q^{-22}+3q^{-26}-2q^{-28}+3q^{-30}+6q^{-36}-6q^{-38}+9q^{-40}-3q^{-42}+q^{-44}+7q^{-46}-7q^{-48}+10q^{-50}-2q^{-52}+4q^{-56}-4q^{-58}+3q^{-60}+q^{-62}-4q^{-64}+2q^{-66}-2q^{-68}-q^{-70}-7q^{-74}+3q^{-76}-6q^{-78}+2q^{-80}-3q^{-82}-6q^{-84}+5q^{-86}-8q^{-88}+5q^{-90}-5q^{-92}-2q^{-94}+4q^{-96}-6q^{-98}+3q^{-100}-q^{-104}+3q^{-106}-2q^{-110}+4q^{-112}-q^{-114}+q^{-116}+q^{-118}-q^{-120}+3q^{-122}-q^{-124}+2q^{-126}+q^{-130}+q^{-134}-q^{-138}+3q^{-140}-3q^{-142}+3q^{-144}-q^{-146}-q^{-148}+q^{-150}-3q^{-152}+3q^{-154}-2q^{-156}+q^{-158}-2q^{-162}+2q^{-164}-2q^{-166}+2q^{-168}-q^{-170}-q^{-176}+q^{-178}-q^{-180}+q^{-182}$

Further Quantum Invariants[show]

Further quantum knot invariants for 10_46.

A1 Invariants.

Weight	Invariant
1	$q^{-1}+2q^{-5}+q^{-9}-q^{-11}-q^{-13}-q^{-17}+q^{-19}-q^{-21}+q^{-23}$
2	$q^{2}-q^{-2}+2q^{-4}+2q^{-6}-2q^{-8}+q^{-10}+3q^{-12}-q^{-14}-q^{-16}+2q^{-18}-q^{-20}-2q^{-22}-2q^{-28}-q^{-30}+2q^{-32}-q^{-36}+2q^{-38}+q^{-40}-q^{-42}+q^{-46}-2q^{-54}+q^{-56}-q^{-60}+q^{-62}$
3	$q^{9}-q^{5}-q^{3}+2q+3q^{-1}-4q^{-5}-q^{-7}+4q^{-9}+5q^{-11}-2q^{-13}-4q^{-15}-q^{-17}+5q^{-19}+4q^{-21}-2q^{-23}-4q^{-25}+3q^{-29}+2q^{-31}-3q^{-33}-5q^{-35}-q^{-37}+2q^{-39}+2q^{-41}-3q^{-43}-q^{-45}+2q^{-47}+4q^{-49}-2q^{-51}-2q^{-53}+2q^{-55}+5q^{-57}-3q^{-59}-5q^{-61}+q^{-63}+6q^{-65}-5q^{-69}-3q^{-71}+2q^{-73}+5q^{-75}+2q^{-77}-4q^{-79}-7q^{-81}+3q^{-83}+6q^{-85}-q^{-87}-6q^{-89}+5q^{-93}+2q^{-95}-3q^{-97}-q^{-99}+q^{-101}+2q^{-103}-q^{-105}-q^{-107}-q^{-115}+q^{-117}$
4	$q^{20}-q^{16}-q^{14}-q^{12}+3q^{10}+3q^{8}+q^{6}-2q^{4}-7q^{2}-1+4q^{-2}+8q^{-4}+6q^{-6}-6q^{-8}-8q^{-10}-7q^{-12}+3q^{-14}+13q^{-16}+7q^{-18}+q^{-20}-11q^{-22}-11q^{-24}+2q^{-26}+9q^{-28}+14q^{-30}+3q^{-32}-10q^{-34}-11q^{-36}-8q^{-38}+7q^{-40}+13q^{-42}+5q^{-44}-5q^{-46}-15q^{-48}-9q^{-50}+5q^{-52}+13q^{-54}+12q^{-56}-5q^{-58}-15q^{-60}-9q^{-62}+5q^{-64}+18q^{-66}+9q^{-68}-9q^{-70}-16q^{-72}-7q^{-74}+13q^{-76}+13q^{-78}-3q^{-80}-11q^{-82}-8q^{-84}+8q^{-86}+10q^{-88}-5q^{-90}-10q^{-92}-2q^{-94}+13q^{-96}+13q^{-98}-8q^{-100}-18q^{-102}-8q^{-104}+12q^{-106}+21q^{-108}+5q^{-110}-15q^{-112}-19q^{-114}-7q^{-116}+14q^{-118}+20q^{-120}+9q^{-122}-11q^{-124}-25q^{-126}-9q^{-128}+15q^{-130}+24q^{-132}+8q^{-134}-21q^{-136}-21q^{-138}+q^{-140}+20q^{-142}+16q^{-144}-12q^{-146}-16q^{-148}-3q^{-150}+11q^{-152}+11q^{-154}-6q^{-156}-8q^{-158}-3q^{-160}+5q^{-162}+6q^{-164}-3q^{-166}-q^{-168}-q^{-170}+q^{-172}+2q^{-174}-2q^{-176}+q^{-178}-q^{-180}-q^{-186}+q^{-188}$
5	$q^{35}-q^{31}-q^{29}-q^{27}+3q^{23}+4q^{21}+q^{19}-2q^{17}-5q^{15}-7q^{13}-2q^{11}+6q^{9}+11q^{7}+9q^{5}+2q^{3}-10q-16q^{-1}-12q^{-3}+q^{-5}+15q^{-7}+21q^{-9}+14q^{-11}-2q^{-13}-19q^{-15}-26q^{-17}-14q^{-19}+7q^{-21}+23q^{-23}+30q^{-25}+16q^{-27}-9q^{-29}-28q^{-31}-28q^{-33}-13q^{-35}+11q^{-37}+32q^{-39}+31q^{-41}+11q^{-43}-15q^{-45}-33q^{-47}-33q^{-49}-11q^{-51}+19q^{-53}+36q^{-55}+32q^{-57}+8q^{-59}-24q^{-61}-43q^{-63}-35q^{-65}-q^{-67}+35q^{-69}+51q^{-71}+29q^{-73}-13q^{-75}-51q^{-77}-52q^{-79}-14q^{-81}+38q^{-83}+62q^{-85}+38q^{-87}-16q^{-89}-61q^{-91}-59q^{-93}-5q^{-95}+49q^{-97}+65q^{-99}+28q^{-101}-32q^{-103}-66q^{-105}-42q^{-107}+14q^{-109}+55q^{-111}+51q^{-113}+5q^{-115}-42q^{-117}-50q^{-119}-14q^{-121}+27q^{-123}+42q^{-125}+20q^{-127}-15q^{-129}-28q^{-131}-12q^{-133}+11q^{-135}+19q^{-137}-2q^{-139}-21q^{-141}-16q^{-143}+12q^{-145}+35q^{-147}+28q^{-149}-14q^{-151}-51q^{-153}-46q^{-155}+47q^{-159}+62q^{-161}+27q^{-163}-30q^{-165}-63q^{-167}-53q^{-169}-5q^{-171}+44q^{-173}+66q^{-175}+46q^{-177}-10q^{-179}-61q^{-181}-69q^{-183}-28q^{-185}+38q^{-187}+79q^{-189}+55q^{-191}-14q^{-193}-69q^{-195}-66q^{-197}-5q^{-199}+53q^{-201}+62q^{-203}+16q^{-205}-39q^{-207}-52q^{-209}-15q^{-211}+30q^{-213}+38q^{-215}+10q^{-217}-23q^{-219}-32q^{-221}-3q^{-223}+23q^{-225}+21q^{-227}-q^{-229}-18q^{-231}-15q^{-233}+q^{-235}+15q^{-237}+10q^{-239}-3q^{-241}-10q^{-243}-5q^{-245}+2q^{-247}+5q^{-249}+3q^{-251}-q^{-253}-3q^{-255}+q^{-259}+q^{-261}-q^{-267}-q^{-273}+q^{-275}$
6	$q^{54}-q^{50}-q^{48}-q^{46}+4q^{40}+4q^{38}+q^{36}-2q^{34}-5q^{32}-6q^{30}-8q^{28}+q^{26}+8q^{24}+13q^{22}+12q^{20}+6q^{18}-4q^{16}-21q^{14}-21q^{12}-16q^{10}+17q^{6}+31q^{4}+33q^{2}+12-9q^{-2}-34q^{-4}-42q^{-6}-36q^{-8}-6q^{-10}+29q^{-12}+48q^{-14}+55q^{-16}+30q^{-18}-6q^{-20}-48q^{-22}-64q^{-24}-52q^{-26}-21q^{-28}+29q^{-30}+64q^{-32}+75q^{-34}+47q^{-36}+3q^{-38}-44q^{-40}-78q^{-42}-74q^{-44}-39q^{-46}+18q^{-48}+64q^{-50}+87q^{-52}+76q^{-54}+24q^{-56}-37q^{-58}-87q^{-60}-99q^{-62}-70q^{-64}-8q^{-66}+69q^{-68}+116q^{-70}+112q^{-72}+51q^{-74}-37q^{-76}-118q^{-78}-151q^{-80}-102q^{-82}+2q^{-84}+114q^{-86}+169q^{-88}+147q^{-90}+40q^{-92}-103q^{-94}-191q^{-96}-181q^{-98}-68q^{-100}+80q^{-102}+201q^{-104}+210q^{-106}+94q^{-108}-73q^{-110}-203q^{-112}-217q^{-114}-116q^{-116}+65q^{-118}+205q^{-120}+220q^{-122}+113q^{-124}-60q^{-126}-195q^{-128}-222q^{-130}-105q^{-132}+69q^{-134}+191q^{-136}+204q^{-138}+97q^{-140}-70q^{-142}-197q^{-144}-187q^{-146}-66q^{-148}+86q^{-150}+188q^{-152}+175q^{-154}+49q^{-156}-111q^{-158}-187q^{-160}-147q^{-162}-20q^{-164}+118q^{-166}+181q^{-168}+122q^{-170}-17q^{-172}-128q^{-174}-154q^{-176}-84q^{-178}+30q^{-180}+117q^{-182}+114q^{-184}+32q^{-186}-45q^{-188}-75q^{-190}-44q^{-192}+8q^{-194}+39q^{-196}+14q^{-198}-39q^{-200}-53q^{-202}-10q^{-204}+69q^{-206}+113q^{-208}+87q^{-210}-21q^{-212}-137q^{-214}-176q^{-216}-106q^{-218}+45q^{-220}+174q^{-222}+214q^{-224}+127q^{-226}-31q^{-228}-178q^{-230}-230q^{-232}-158q^{-234}-11q^{-236}+151q^{-238}+236q^{-240}+212q^{-242}+63q^{-244}-122q^{-246}-249q^{-248}-251q^{-250}-108q^{-252}+98q^{-254}+267q^{-256}+268q^{-258}+119q^{-260}-100q^{-262}-261q^{-264}-262q^{-266}-108q^{-268}+117q^{-270}+242q^{-272}+218q^{-274}+66q^{-276}-116q^{-278}-209q^{-280}-162q^{-282}-10q^{-284}+114q^{-286}+153q^{-288}+90q^{-290}-19q^{-292}-97q^{-294}-94q^{-296}-22q^{-298}+32q^{-300}+54q^{-302}+34q^{-304}-3q^{-306}-25q^{-308}-19q^{-310}+10q^{-312}+8q^{-314}-q^{-316}-9q^{-318}-11q^{-320}-q^{-322}+10q^{-324}+21q^{-326}+3q^{-328}-12q^{-330}-14q^{-332}-7q^{-334}+4q^{-336}+9q^{-338}+11q^{-340}-2q^{-342}-7q^{-344}-5q^{-346}-2q^{-348}+4q^{-350}+2q^{-352}+3q^{-354}-2q^{-356}-2q^{-358}+3q^{-364}-q^{-366}-q^{-370}-q^{-376}+q^{-378}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-12}+q^{-14}-2q^{-16}-q^{-18}-3q^{-20}-q^{-22}+q^{-28}+q^{-32}$
1,1	$q^{-4}+6q^{-8}-4q^{-10}+14q^{-12}-12q^{-14}+24q^{-16}-18q^{-18}+20q^{-20}-16q^{-22}+6q^{-24}-2q^{-26}-14q^{-28}+10q^{-30}-26q^{-32}+22q^{-34}-29q^{-36}+26q^{-38}-22q^{-40}+22q^{-42}-9q^{-44}+10q^{-46}+5q^{-52}-2q^{-54}-2q^{-62}-6q^{-66}+5q^{-68}-6q^{-70}+8q^{-72}-6q^{-74}+5q^{-76}-4q^{-78}+4q^{-80}-4q^{-82}+3q^{-84}-2q^{-86}+2q^{-88}-2q^{-90}+q^{-92}$
2,0	$q^{-4}+q^{-6}+q^{-8}+q^{-10}+3q^{-12}+3q^{-14}+2q^{-16}+q^{-18}+3q^{-20}+q^{-22}-q^{-24}-2q^{-26}-q^{-28}-3q^{-30}-4q^{-32}-4q^{-34}-4q^{-36}-4q^{-38}-2q^{-40}+q^{-42}+2q^{-44}+4q^{-46}+6q^{-48}+5q^{-50}+q^{-52}+q^{-54}-2q^{-60}-q^{-62}-q^{-64}-q^{-66}-2q^{-68}-q^{-70}+q^{-76}+q^{-80}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-8}+3q^{-12}+3q^{-14}+4q^{-16}+6q^{-18}+5q^{-20}+2q^{-22}+q^{-24}-4q^{-26}-8q^{-28}-8q^{-30}-8q^{-32}-5q^{-34}+q^{-38}+3q^{-40}+5q^{-42}+3q^{-44}+2q^{-46}+q^{-48}+2q^{-50}-q^{-54}-2q^{-60}+q^{-64}-2q^{-66}+2q^{-70}-q^{-72}-q^{-74}+q^{-76}$
1,0,0	$q^{-7}+q^{-9}+3q^{-11}+2q^{-13}+4q^{-15}+q^{-17}+q^{-19}-2q^{-21}-3q^{-23}-4q^{-25}-3q^{-27}-q^{-29}-q^{-31}+2q^{-33}+2q^{-37}+q^{-41}$
1,0,1	$q^{-10}+6q^{-14}+2q^{-16}+13q^{-18}+8q^{-20}+14q^{-22}+14q^{-24}+6q^{-26}+12q^{-28}-10q^{-30}-27q^{-34}-13q^{-36}-30q^{-38}-17q^{-40}-13q^{-42}-14q^{-44}+14q^{-46}-8q^{-48}+32q^{-50}+q^{-52}+27q^{-54}+5q^{-56}+7q^{-58}+9q^{-60}-10q^{-62}+7q^{-64}-17q^{-66}+8q^{-68}-13q^{-70}+4q^{-72}-2q^{-74}-3q^{-76}+5q^{-78}-4q^{-80}+5q^{-82}-4q^{-84}+2q^{-86}-3q^{-88}-q^{-90}-q^{-92}-q^{-94}+4q^{-96}-q^{-98}+q^{-100}-q^{-104}+2q^{-106}-q^{-108}-q^{-110}+2q^{-112}-2q^{-114}+q^{-116}+q^{-118}-2q^{-120}+q^{-122}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-14}+q^{-16}+3q^{-18}+5q^{-20}+7q^{-22}+9q^{-24}+11q^{-26}+10q^{-28}+8q^{-30}+3q^{-32}-3q^{-34}-10q^{-36}-16q^{-38}-21q^{-40}-19q^{-42}-16q^{-44}-9q^{-46}-3q^{-48}+5q^{-50}+12q^{-52}+12q^{-54}+11q^{-56}+11q^{-58}+9q^{-60}+2q^{-62}+2q^{-64}-3q^{-68}-4q^{-70}-2q^{-72}-3q^{-74}-4q^{-76}-q^{-78}+q^{-80}-q^{-82}-q^{-84}+2q^{-86}+q^{-88}-q^{-90}+q^{-94}$
1,0,0,0	$q^{-10}+q^{-12}+3q^{-14}+3q^{-16}+4q^{-18}+4q^{-20}+2q^{-22}+q^{-24}-3q^{-26}-3q^{-28}-6q^{-30}-4q^{-32}-4q^{-34}-q^{-36}+q^{-40}+2q^{-42}+q^{-44}+2q^{-46}+q^{-50}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-8}+3q^{-12}-q^{-14}+4q^{-16}-2q^{-18}+5q^{-20}-2q^{-22}+3q^{-24}-2q^{-26}-4q^{-32}+3q^{-34}-6q^{-36}+5q^{-38}-7q^{-40}+5q^{-42}-5q^{-44}+4q^{-46}-3q^{-48}+2q^{-50}-q^{-54}+2q^{-56}-2q^{-58}+2q^{-60}-2q^{-62}+3q^{-64}-2q^{-66}+2q^{-68}-2q^{-70}+q^{-72}-q^{-74}+q^{-76}$
1,0	$q^{-10}+3q^{-18}+2q^{-20}+4q^{-26}+4q^{-28}+2q^{-30}-2q^{-32}+q^{-34}+3q^{-36}+2q^{-38}-3q^{-40}-5q^{-42}-2q^{-44}-2q^{-48}-6q^{-50}-5q^{-52}-q^{-54}+q^{-56}-q^{-58}-q^{-60}+3q^{-64}+q^{-66}+q^{-70}+4q^{-72}+2q^{-74}-q^{-76}-q^{-78}+2q^{-80}+3q^{-82}-2q^{-86}-q^{-88}+q^{-90}+q^{-92}-q^{-94}-2q^{-96}-q^{-98}+q^{-100}+2q^{-102}-q^{-104}-2q^{-106}-q^{-108}+q^{-110}+2q^{-112}-q^{-116}-q^{-118}+q^{-122}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-14}+3q^{-18}+q^{-20}+7q^{-22}+3q^{-24}+9q^{-26}+4q^{-28}+9q^{-30}+q^{-32}+2q^{-34}-5q^{-36}-6q^{-38}-10q^{-40}-12q^{-42}-8q^{-44}-10q^{-46}-q^{-48}-6q^{-50}+6q^{-52}+10q^{-56}+8q^{-60}-q^{-62}+5q^{-64}-q^{-66}+2q^{-68}-q^{-70}+q^{-74}-q^{-76}+q^{-78}-2q^{-80}+q^{-82}-2q^{-84}+q^{-86}-2q^{-88}+2q^{-90}-2q^{-92}+q^{-94}-q^{-96}+2q^{-98}-q^{-100}-q^{-104}+q^{-106}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-22}+3q^{-26}-2q^{-28}+3q^{-30}+6q^{-36}-6q^{-38}+9q^{-40}-3q^{-42}+q^{-44}+7q^{-46}-7q^{-48}+10q^{-50}-2q^{-52}+4q^{-56}-4q^{-58}+3q^{-60}+q^{-62}-4q^{-64}+2q^{-66}-2q^{-68}-q^{-70}-7q^{-74}+3q^{-76}-6q^{-78}+2q^{-80}-3q^{-82}-6q^{-84}+5q^{-86}-8q^{-88}+5q^{-90}-5q^{-92}-2q^{-94}+4q^{-96}-6q^{-98}+3q^{-100}-q^{-104}+3q^{-106}-2q^{-110}+4q^{-112}-q^{-114}+q^{-116}+q^{-118}-q^{-120}+3q^{-122}-q^{-124}+2q^{-126}+q^{-130}+q^{-134}-q^{-138}+3q^{-140}-3q^{-142}+3q^{-144}-q^{-146}-q^{-148}+q^{-150}-3q^{-152}+3q^{-154}-2q^{-156}+q^{-158}-2q^{-162}+2q^{-164}-2q^{-166}+2q^{-168}-q^{-170}-q^{-176}+q^{-178}-q^{-180}+q^{-182}$

.

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 46"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{4}+3t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-6z^{4}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, 6 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{11}-2q^{10}+3q^{9}-4q^{8}+4q^{7}-5q^{6}+4q^{5}-3q^{4}+3q^{3}-q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+5z^{4}a^{-8}+11z^{2}a^{-4}-18z^{2}a^{-6}+7z^{2}a^{-8}+6a^{-4}-8a^{-6}+3a^{-8}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-5z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-4}-23z^{6}a^{-6}-12z^{6}a^{-8}+4z^{6}a^{-10}+5z^{5}a^{-5}-12z^{5}a^{-7}-13z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-4}+42z^{4}a^{-6}+13z^{4}a^{-8}-9z^{4}a^{-10}+3z^{4}a^{-12}+5z^{3}a^{-5}+23z^{3}a^{-7}+9z^{3}a^{-9}-7z^{3}a^{-11}+2z^{3}a^{-13}-17z^{2}a^{-4}-29z^{2}a^{-6}-7z^{2}a^{-8}+2z^{2}a^{-10}-2z^{2}a^{-12}+z^{2}a^{-14}-6za^{-5}-10za^{-7}-2za^{-9}+2za^{-11}+6a^{-4}+8a^{-6}+3a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n60,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 46"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{4}+3t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$, $q^{11}-2q^{10}+3q^{9}-4q^{8}+4q^{7}-5q^{6}+4q^{5}-3q^{4}+3q^{3}-q^{2}+q$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n60,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(0, -4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $-32$ $0$ $-96$ $48$ $0$ ${\frac {256}{3}}$ ${\frac {544}{3}}$ $256$ $0$ $512$ $0$ $0$ $2224$ $-176$ $2192$ $304$ $112$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$6 is the signature of 10 46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   1   -1 19                 2 1   1 17               2 1     -1 15             2 2       0 13           3 2         -1 11         1 2           -1 9       2 3             1 7     1 1               0 5   1 3                 2 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)[show]

$n$	$J_{n}$
2	$q^{30}-2q^{29}+q^{28}+2q^{27}-5q^{26}+3q^{25}+2q^{24}-5q^{23}+4q^{22}+q^{21}-6q^{20}+6q^{19}+2q^{18}-9q^{17}+7q^{16}+4q^{15}-12q^{14}+6q^{13}+6q^{12}-12q^{11}+4q^{10}+7q^{9}-9q^{8}+q^{7}+7q^{6}-5q^{5}-q^{4}+4q^{3}-q^{2}-q+1$
3	$q^{57}-2q^{56}+q^{55}+q^{53}-3q^{52}+q^{51}+3q^{50}-5q^{48}-q^{47}+8q^{46}+3q^{45}-10q^{44}-7q^{43}+13q^{42}+10q^{41}-13q^{40}-17q^{39}+16q^{38}+16q^{37}-10q^{36}-20q^{35}+11q^{34}+14q^{33}-5q^{32}-14q^{31}+6q^{30}+8q^{29}-3q^{28}-6q^{27}+3q^{26}+4q^{25}-3q^{24}+q^{22}+q^{21}-5q^{20}+5q^{19}+q^{18}-2q^{17}-9q^{16}+7q^{15}+6q^{14}-q^{13}-12q^{12}+3q^{11}+8q^{10}+5q^{9}-11q^{8}-3q^{7}+5q^{6}+7q^{5}-4q^{4}-4q^{3}+4q-q^{-1}-q^{-2}+q^{-3}$
4	$q^{92}-2q^{91}+q^{90}-q^{88}+3q^{87}-5q^{86}+5q^{85}-q^{84}-3q^{83}+3q^{82}-7q^{81}+14q^{80}-2q^{79}-11q^{78}-2q^{77}-5q^{76}+31q^{75}-2q^{74}-25q^{73}-15q^{72}-q^{71}+59q^{70}+2q^{69}-44q^{68}-37q^{67}-q^{66}+88q^{65}+18q^{64}-53q^{63}-61q^{62}-17q^{61}+102q^{60}+38q^{59}-42q^{58}-67q^{57}-38q^{56}+90q^{55}+42q^{54}-22q^{53}-51q^{52}-47q^{51}+70q^{50}+32q^{49}-12q^{48}-30q^{47}-47q^{46}+55q^{45}+24q^{44}-7q^{43}-15q^{42}-49q^{41}+39q^{40}+21q^{39}+q^{38}+q^{37}-49q^{36}+19q^{35}+12q^{34}+8q^{33}+19q^{32}-40q^{31}+6q^{30}-2q^{29}+2q^{28}+29q^{27}-23q^{26}+7q^{25}-10q^{24}-12q^{23}+23q^{22}-13q^{21}+17q^{20}-2q^{19}-18q^{18}+8q^{17}-16q^{16}+18q^{15}+11q^{14}-7q^{13}+3q^{12}-23q^{11}+5q^{10}+11q^{9}+5q^{8}+9q^{7}-17q^{6}-5q^{5}+q^{4}+4q^{3}+11q^{2}-5q-3-3q^{-1}-q^{-2}+5q^{-3}-q^{-6}-q^{-7}+q^{-8}$
5	$q^{135}-2q^{134}+q^{133}-q^{131}+q^{130}+q^{129}-q^{128}+q^{127}-q^{126}-4q^{125}+3q^{124}+5q^{123}+q^{122}-2q^{121}-8q^{120}-9q^{119}+10q^{118}+18q^{117}+6q^{116}-16q^{115}-24q^{114}-12q^{113}+27q^{112}+40q^{111}+8q^{110}-42q^{109}-53q^{108}-3q^{107}+60q^{106}+68q^{105}-87q^{103}-90q^{102}+10q^{101}+115q^{100}+114q^{99}-9q^{98}-145q^{97}-151q^{96}+7q^{95}+170q^{94}+183q^{93}+15q^{92}-186q^{91}-217q^{90}-34q^{89}+178q^{88}+234q^{87}+71q^{86}-166q^{85}-239q^{84}-83q^{83}+130q^{82}+224q^{81}+104q^{80}-109q^{79}-204q^{78}-98q^{77}+83q^{76}+178q^{75}+99q^{74}-72q^{73}-162q^{72}-91q^{71}+60q^{70}+150q^{69}+94q^{68}-53q^{67}-141q^{66}-99q^{65}+37q^{64}+134q^{63}+107q^{62}-18q^{61}-119q^{60}-114q^{59}-4q^{58}+98q^{57}+115q^{56}+29q^{55}-73q^{54}-110q^{53}-45q^{52}+42q^{51}+91q^{50}+63q^{49}-13q^{48}-73q^{47}-61q^{46}-12q^{45}+37q^{44}+61q^{43}+32q^{42}-19q^{41}-37q^{40}-36q^{39}-15q^{38}+23q^{37}+33q^{36}+19q^{35}+5q^{34}-14q^{33}-31q^{32}-13q^{31}-q^{30}+11q^{29}+24q^{28}+18q^{27}-7q^{26}-9q^{25}-18q^{24}-19q^{23}+2q^{22}+18q^{21}+11q^{20}+17q^{19}+2q^{18}-18q^{17}-19q^{16}-6q^{15}-4q^{14}+17q^{13}+21q^{12}+7q^{11}-5q^{10}-13q^{9}-20q^{8}-4q^{7}+9q^{6}+14q^{5}+12q^{4}+3q^{3}-13q^{2}-10q-5+q^{-1}+8q^{-2}+9q^{-3}-q^{-4}-3q^{-5}-3q^{-6}-4q^{-7}+4q^{-9}+q^{-10}-q^{-13}-q^{-14}+q^{-15}$
6	$q^{186}-2q^{185}+q^{184}-q^{182}+q^{181}-q^{180}+5q^{179}-5q^{178}+q^{177}-2q^{176}-q^{175}+6q^{174}-2q^{173}+7q^{172}-11q^{171}-2q^{170}-4q^{169}+4q^{168}+19q^{167}-4q^{166}+2q^{165}-22q^{164}-9q^{163}-2q^{162}+19q^{161}+37q^{160}-15q^{159}-9q^{158}-32q^{157}-7q^{156}+6q^{155}+28q^{154}+39q^{153}-44q^{152}-15q^{151}-10q^{150}+30q^{149}+26q^{148}+6q^{147}-15q^{146}-116q^{145}-18q^{144}+68q^{143}+139q^{142}+89q^{141}-33q^{140}-139q^{139}-268q^{138}-65q^{137}+161q^{136}+321q^{135}+241q^{134}-9q^{133}-264q^{132}-493q^{131}-219q^{130}+162q^{129}+482q^{128}+460q^{127}+140q^{126}-265q^{125}-662q^{124}-425q^{123}+19q^{122}+484q^{121}+587q^{120}+325q^{119}-116q^{118}-638q^{117}-510q^{116}-143q^{115}+337q^{114}+515q^{113}+377q^{112}+31q^{111}-480q^{110}-423q^{109}-183q^{108}+208q^{107}+364q^{106}+307q^{105}+70q^{104}-364q^{103}-315q^{102}-157q^{101}+164q^{100}+285q^{99}+264q^{98}+84q^{97}-311q^{96}-290q^{95}-188q^{94}+112q^{93}+254q^{92}+294q^{91}+161q^{90}-229q^{89}-287q^{88}-275q^{87}-2q^{86}+184q^{85}+320q^{84}+272q^{83}-90q^{82}-228q^{81}-338q^{80}-140q^{79}+57q^{78}+280q^{77}+348q^{76}+70q^{75}-102q^{74}-325q^{73}-242q^{72}-95q^{71}+159q^{70}+338q^{69}+197q^{68}+65q^{67}-218q^{66}-255q^{65}-217q^{64}-15q^{63}+221q^{62}+224q^{61}+200q^{60}-45q^{59}-148q^{58}-232q^{57}-155q^{56}+40q^{55}+123q^{54}+214q^{53}+85q^{52}+19q^{51}-116q^{50}-164q^{49}-81q^{48}-25q^{47}+101q^{46}+75q^{45}+107q^{44}+27q^{43}-57q^{42}-59q^{41}-80q^{40}-11q^{39}-29q^{38}+58q^{37}+60q^{36}+24q^{35}+29q^{34}-19q^{33}-7q^{32}-76q^{31}-19q^{30}-2q^{29}-5q^{28}+41q^{27}+31q^{26}+54q^{25}-24q^{24}-8q^{23}-25q^{22}-51q^{21}-16q^{20}-4q^{19}+50q^{18}+10q^{17}+39q^{16}+19q^{15}-23q^{14}-27q^{13}-39q^{12}-21q^{10}+27q^{9}+35q^{8}+19q^{7}+9q^{6}-14q^{5}-7q^{4}-40q^{3}-8q^{2}+5q+13+17q^{-1}+11q^{-2}+14q^{-3}-19q^{-4}-10q^{-5}-9q^{-6}-4q^{-7}+q^{-8}+6q^{-9}+14q^{-10}-2q^{-11}-3q^{-13}-3q^{-14}-4q^{-15}-q^{-16}+5q^{-17}+q^{-19}-q^{-22}-q^{-23}+q^{-24}$
7	$q^{245}-2q^{244}+q^{243}-q^{241}+q^{240}-q^{239}+3q^{238}+q^{237}-5q^{236}+q^{234}+2q^{233}+2q^{232}-5q^{231}+4q^{230}+q^{229}-9q^{228}-q^{227}+7q^{226}+8q^{225}+3q^{224}-11q^{223}-2q^{222}-4q^{221}-10q^{220}+5q^{219}+13q^{218}+17q^{217}+9q^{216}-13q^{215}-17q^{214}-21q^{213}-18q^{212}+6q^{211}+19q^{210}+37q^{209}+43q^{208}+14q^{207}-21q^{206}-66q^{205}-85q^{204}-51q^{203}+8q^{202}+100q^{201}+161q^{200}+135q^{199}+17q^{198}-155q^{197}-282q^{196}-250q^{195}-62q^{194}+214q^{193}+433q^{192}+424q^{191}+156q^{190}-272q^{189}-637q^{188}-658q^{187}-289q^{186}+315q^{185}+845q^{184}+942q^{183}+498q^{182}-294q^{181}-1049q^{180}-1273q^{179}-787q^{178}+195q^{177}+1210q^{176}+1602q^{175}+1130q^{174}+4q^{173}-1247q^{172}-1881q^{171}-1523q^{170}-310q^{169}+1183q^{168}+2065q^{167}+1849q^{166}+655q^{165}-965q^{164}-2080q^{163}-2090q^{162}-998q^{161}+686q^{160}+1958q^{159}+2159q^{158}+1222q^{157}-386q^{156}-1716q^{155}-2067q^{154}-1312q^{153}+162q^{152}+1450q^{151}+1856q^{150}+1248q^{149}-51q^{148}-1220q^{147}-1617q^{146}-1094q^{145}+60q^{144}+1084q^{143}+1401q^{142}+913q^{141}-112q^{140}-1023q^{139}-1285q^{138}-793q^{137}+190q^{136}+1018q^{135}+1213q^{134}+730q^{133}-182q^{132}-983q^{131}-1207q^{130}-761q^{129}+132q^{128}+926q^{127}+1171q^{126}+800q^{125}-3q^{124}-779q^{123}-1116q^{122}-863q^{121}-152q^{120}+610q^{119}+1011q^{118}+881q^{117}+307q^{116}-388q^{115}-856q^{114}-881q^{113}-461q^{112}+169q^{111}+683q^{110}+832q^{109}+569q^{108}+61q^{107}-468q^{106}-741q^{105}-664q^{104}-278q^{103}+243q^{102}+617q^{101}+690q^{100}+458q^{99}+9q^{98}-421q^{97}-669q^{96}-620q^{95}-244q^{94}+210q^{93}+559q^{92}+675q^{91}+464q^{90}+65q^{89}-386q^{88}-678q^{87}-608q^{86}-299q^{85}+141q^{84}+547q^{83}+671q^{82}+517q^{81}+120q^{80}-358q^{79}-605q^{78}-627q^{77}-368q^{76}+92q^{75}+455q^{74}+635q^{73}+526q^{72}+152q^{71}-204q^{70}-510q^{69}-593q^{68}-358q^{67}-36q^{66}+315q^{65}+510q^{64}+432q^{63}+259q^{62}-61q^{61}-353q^{60}-416q^{59}-357q^{58}-129q^{57}+136q^{56}+262q^{55}+356q^{54}+261q^{53}+43q^{52}-96q^{51}-247q^{50}-252q^{49}-150q^{48}-69q^{47}+97q^{46}+176q^{45}+147q^{44}+150q^{43}+33q^{42}-55q^{41}-78q^{40}-141q^{39}-85q^{38}-43q^{37}-31q^{36}+76q^{35}+78q^{34}+71q^{33}+83q^{32}+10q^{31}-q^{30}-36q^{29}-109q^{28}-55q^{27}-48q^{26}-21q^{25}+48q^{24}+44q^{23}+83q^{22}+81q^{21}-q^{20}-7q^{19}-51q^{18}-80q^{17}-44q^{16}-50q^{15}+5q^{14}+59q^{13}+42q^{12}+67q^{11}+36q^{10}-8q^{9}-13q^{8}-55q^{7}-55q^{6}-23q^{5}-22q^{4}+27q^{3}+40q^{2}+28q+40+9q^{-1}-13q^{-2}-20q^{-3}-42q^{-4}-19q^{-5}-3q^{-6}-2q^{-7}+21q^{-8}+21q^{-9}+17q^{-10}+13q^{-11}-13q^{-12}-11q^{-13}-8q^{-14}-13q^{-15}-4q^{-16}+q^{-17}+7q^{-18}+13q^{-19}+q^{-20}+q^{-22}-4q^{-23}-3q^{-24}-4q^{-25}-q^{-26}+4q^{-27}+q^{-28}+q^{-30}-q^{-33}-q^{-34}+q^{-35}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

10_45 10_47