9 10

9_9

9_11

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 10 at Knotilus!

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13
Gauss code	1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 6 4 10
Conway Notation	[333]

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

[edit Notes on presentations of 9 10]

Computer Talk

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["9 10"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 12 14 16 18 2 6 4 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[333]

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]=

${\textrm {BR}}(4,\{1,1,2,-1,2,2,2,2,3,-2,3\})$

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]=

{ 4, 11, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [3][-14] Hyperbolic Volume 8.77346 A-Polynomial See Data:9 10/A-polynomial

[edit Notes for 9 10's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for 9 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-8t+9-8t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$-q^{11}+q^{10}-3q^{9}+5q^{8}-5q^{7}+6q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-6}-z^{6}a^{-8}-3z^{6}a^{-10}+z^{6}a^{-12}+2z^{5}a^{-5}-3z^{5}a^{-7}-7z^{5}a^{-9}-z^{5}a^{-11}+z^{5}a^{-13}+z^{4}a^{-4}-7z^{4}a^{-6}+3z^{4}a^{-8}+9z^{4}a^{-10}-2z^{4}a^{-12}-3z^{3}a^{-5}+3z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}-4z^{3}a^{-13}-2z^{2}a^{-4}+7z^{2}a^{-6}-2z^{2}a^{-8}-11z^{2}a^{-10}-4za^{-9}+4za^{-13}-2a^{-6}+a^{-8}+2a^{-10}$
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-9q^{-46}+11q^{-48}-8q^{-50}+3q^{-52}+5q^{-54}-13q^{-56}+21q^{-58}-19q^{-60}+12q^{-62}-2q^{-64}-10q^{-66}+18q^{-68}-17q^{-70}+14q^{-72}-2q^{-74}-7q^{-76}+13q^{-78}-9q^{-80}-q^{-82}+12q^{-84}-19q^{-86}+18q^{-88}-9q^{-90}-4q^{-92}+21q^{-94}-28q^{-96}+31q^{-98}-20q^{-100}+6q^{-102}+11q^{-104}-21q^{-106}+26q^{-108}-18q^{-110}+12q^{-112}+2q^{-114}-10q^{-116}+15q^{-118}-10q^{-120}-2q^{-122}+9q^{-124}-16q^{-126}+10q^{-128}-3q^{-130}-12q^{-132}+18q^{-134}-20q^{-136}+15q^{-138}-10q^{-140}-9q^{-142}+13q^{-144}-16q^{-146}+13q^{-148}-9q^{-150}+2q^{-152}+3q^{-154}-4q^{-156}+6q^{-158}-6q^{-160}+4q^{-162}-q^{-164}+q^{-168}-2q^{-170}+2q^{-172}+q^{-176}$

Further Quantum Invariants

Further quantum knot invariants for 9_10.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-q^{-5}+2q^{-7}-q^{-9}+q^{-11}+q^{-13}+2q^{-17}-2q^{-19}-q^{-23}$
2	$q^{-6}-q^{-8}+4q^{-12}-2q^{-14}-3q^{-16}+7q^{-18}-q^{-20}-6q^{-22}+7q^{-24}+q^{-26}-5q^{-28}+2q^{-30}+2q^{-32}-3q^{-36}+4q^{-38}+3q^{-40}-7q^{-42}+q^{-44}+5q^{-46}-7q^{-48}-2q^{-50}+4q^{-52}-3q^{-54}-q^{-56}+2q^{-58}+q^{-64}$
3	$q^{-9}-q^{-11}+2q^{-15}+2q^{-17}-2q^{-19}-3q^{-21}+4q^{-23}+7q^{-25}-4q^{-27}-10q^{-29}+2q^{-31}+17q^{-33}+2q^{-35}-20q^{-37}-6q^{-39}+21q^{-41}+12q^{-43}-20q^{-45}-14q^{-47}+17q^{-49}+15q^{-51}-9q^{-53}-14q^{-55}+3q^{-57}+9q^{-59}+3q^{-61}-9q^{-63}-9q^{-65}+6q^{-67}+15q^{-69}-2q^{-71}-20q^{-73}+q^{-75}+20q^{-77}+3q^{-79}-23q^{-81}-9q^{-83}+16q^{-85}+13q^{-87}-16q^{-89}-13q^{-91}+8q^{-93}+14q^{-95}-2q^{-97}-10q^{-99}+q^{-101}+7q^{-103}+2q^{-105}-3q^{-107}+q^{-111}+q^{-113}-q^{-115}-q^{-123}$
4	$q^{-12}-q^{-14}+2q^{-18}+2q^{-22}-3q^{-24}-q^{-26}+5q^{-28}+5q^{-32}-8q^{-34}-6q^{-36}+9q^{-38}+6q^{-40}+14q^{-42}-17q^{-44}-24q^{-46}+q^{-48}+19q^{-50}+47q^{-52}-10q^{-54}-51q^{-56}-33q^{-58}+14q^{-60}+83q^{-62}+23q^{-64}-54q^{-66}-70q^{-68}-17q^{-70}+90q^{-72}+55q^{-74}-28q^{-76}-73q^{-78}-42q^{-80}+56q^{-82}+55q^{-84}+6q^{-86}-43q^{-88}-44q^{-90}+11q^{-92}+35q^{-94}+26q^{-96}-10q^{-98}-34q^{-100}-28q^{-102}+15q^{-104}+44q^{-106}+17q^{-108}-28q^{-110}-58q^{-112}+60q^{-116}+42q^{-118}-19q^{-120}-84q^{-122}-19q^{-124}+59q^{-126}+62q^{-128}+5q^{-130}-86q^{-132}-45q^{-134}+27q^{-136}+66q^{-138}+43q^{-140}-53q^{-142}-52q^{-144}-10q^{-146}+38q^{-148}+55q^{-150}-8q^{-152}-29q^{-154}-28q^{-156}+2q^{-158}+32q^{-160}+9q^{-162}-4q^{-164}-16q^{-166}-9q^{-168}+8q^{-170}+3q^{-172}+4q^{-174}-3q^{-176}-4q^{-178}+q^{-180}-2q^{-182}+2q^{-184}-q^{-188}+q^{-190}-q^{-192}+q^{-200}$
5	$q^{-15}-q^{-17}+2q^{-21}+q^{-27}-q^{-29}-q^{-31}+3q^{-33}+3q^{-35}-2q^{-37}-q^{-41}+4q^{-45}+6q^{-47}-q^{-49}-9q^{-51}-11q^{-53}-4q^{-55}+14q^{-57}+26q^{-59}+22q^{-61}-9q^{-63}-49q^{-65}-52q^{-67}-8q^{-69}+62q^{-71}+99q^{-73}+56q^{-75}-60q^{-77}-153q^{-79}-118q^{-81}+31q^{-83}+186q^{-85}+199q^{-87}+30q^{-89}-197q^{-91}-273q^{-93}-104q^{-95}+171q^{-97}+314q^{-99}+183q^{-101}-115q^{-103}-323q^{-105}-243q^{-107}+53q^{-109}+287q^{-111}+266q^{-113}+15q^{-115}-227q^{-117}-259q^{-119}-71q^{-121}+158q^{-123}+220q^{-125}+96q^{-127}-80q^{-129}-173q^{-131}-115q^{-133}+26q^{-135}+121q^{-137}+115q^{-139}+26q^{-141}-78q^{-143}-122q^{-145}-65q^{-147}+45q^{-149}+124q^{-151}+100q^{-153}-22q^{-155}-140q^{-157}-139q^{-159}+5q^{-161}+164q^{-163}+177q^{-165}+17q^{-167}-179q^{-169}-224q^{-171}-48q^{-173}+193q^{-175}+264q^{-177}+89q^{-179}-176q^{-181}-302q^{-183}-140q^{-185}+144q^{-187}+300q^{-189}+204q^{-191}-76q^{-193}-287q^{-195}-239q^{-197}+2q^{-199}+221q^{-201}+257q^{-203}+86q^{-205}-151q^{-207}-235q^{-209}-128q^{-211}+58q^{-213}+184q^{-215}+157q^{-217}+16q^{-219}-118q^{-221}-140q^{-223}-62q^{-225}+48q^{-227}+102q^{-229}+76q^{-231}-3q^{-233}-64q^{-235}-64q^{-237}-23q^{-239}+22q^{-241}+44q^{-243}+25q^{-245}-3q^{-247}-20q^{-249}-20q^{-251}-7q^{-253}+9q^{-255}+11q^{-257}+6q^{-259}+2q^{-261}-5q^{-263}-6q^{-265}+q^{-269}+q^{-271}+4q^{-273}-2q^{-277}-q^{-283}+q^{-285}+q^{-287}-q^{-295}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
1,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+17q^{-20}-22q^{-22}+32q^{-24}-42q^{-26}+58q^{-28}-56q^{-30}+60q^{-32}-54q^{-34}+43q^{-36}-16q^{-38}-12q^{-40}+44q^{-42}-69q^{-44}+100q^{-46}-114q^{-48}+118q^{-50}-119q^{-52}+94q^{-54}-80q^{-56}+46q^{-58}-24q^{-60}-8q^{-62}+36q^{-64}-46q^{-66}+56q^{-68}-56q^{-70}+56q^{-72}-44q^{-74}+30q^{-76}-28q^{-78}+16q^{-80}-12q^{-82}+8q^{-84}-4q^{-86}+4q^{-88}+q^{-92}$
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+2q^{-20}-3q^{-22}-q^{-24}+4q^{-26}+3q^{-28}-2q^{-30}+q^{-32}+4q^{-34}-q^{-36}-2q^{-38}+2q^{-40}+q^{-42}+4q^{-46}+4q^{-48}+2q^{-54}-6q^{-58}-q^{-60}-3q^{-64}-6q^{-66}-3q^{-68}-q^{-72}-q^{-74}+2q^{-78}+q^{-80}+2q^{-82}+q^{-84}+q^{-86}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+q^{-22}+6q^{-24}-4q^{-26}+q^{-28}+7q^{-30}-2q^{-32}+q^{-34}+6q^{-36}+q^{-38}+2q^{-44}-2q^{-46}-4q^{-48}+3q^{-50}-8q^{-54}+2q^{-56}-q^{-58}-7q^{-60}+q^{-62}-q^{-66}+2q^{-68}+2q^{-70}+q^{-74}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+2q^{-21}+q^{-23}+q^{-25}+2q^{-27}+q^{-29}+2q^{-31}+q^{-35}-2q^{-37}-q^{-39}-2q^{-41}-q^{-43}-q^{-45}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+3q^{-30}+2q^{-32}-2q^{-34}+q^{-36}+5q^{-38}+2q^{-40}-2q^{-42}+4q^{-44}+8q^{-46}+q^{-50}+6q^{-52}+3q^{-54}-q^{-56}+4q^{-58}+5q^{-60}-q^{-64}+2q^{-66}-4q^{-68}-11q^{-70}-7q^{-72}-5q^{-74}-10q^{-76}-8q^{-78}+2q^{-82}+q^{-84}+2q^{-86}+5q^{-88}+3q^{-90}+q^{-92}+q^{-94}+q^{-96}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+2q^{-26}+q^{-28}+2q^{-30}+q^{-32}+2q^{-34}+q^{-36}+2q^{-38}+q^{-40}+q^{-44}-2q^{-46}-q^{-48}-2q^{-50}-2q^{-52}-q^{-54}-q^{-56}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-5q^{-22}+6q^{-24}-4q^{-26}+5q^{-28}-q^{-30}+5q^{-34}-6q^{-36}+9q^{-38}-10q^{-40}+10q^{-42}-10q^{-44}+8q^{-46}-6q^{-48}+3q^{-50}-2q^{-54}+4q^{-56}-5q^{-58}+5q^{-60}-5q^{-62}+4q^{-64}-3q^{-66}+2q^{-68}-2q^{-70}-q^{-74}$
1,0	$q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-2q^{-34}+4q^{-36}+6q^{-38}-q^{-40}-5q^{-42}-2q^{-44}+6q^{-46}+5q^{-48}-2q^{-50}-4q^{-52}+3q^{-54}+5q^{-56}+q^{-58}-3q^{-60}+4q^{-64}+2q^{-66}-3q^{-68}-3q^{-70}+2q^{-72}+3q^{-74}-2q^{-76}-4q^{-78}+q^{-80}+5q^{-82}-6q^{-86}-5q^{-88}+3q^{-90}+4q^{-92}-3q^{-94}-7q^{-96}-3q^{-98}+3q^{-100}+2q^{-102}-q^{-104}-2q^{-106}+2q^{-110}+2q^{-112}+q^{-120}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-3q^{-32}+6q^{-34}-3q^{-36}+4q^{-38}-q^{-40}+5q^{-42}+2q^{-44}-q^{-46}+5q^{-48}-2q^{-50}+9q^{-52}-5q^{-54}+8q^{-56}-7q^{-58}+9q^{-60}-7q^{-62}+5q^{-64}-8q^{-66}+2q^{-68}-3q^{-70}-q^{-72}-2q^{-74}-4q^{-76}+2q^{-78}-5q^{-80}+2q^{-82}-6q^{-84}+3q^{-86}-4q^{-88}+2q^{-90}-2q^{-92}+3q^{-94}+2q^{-98}+q^{-102}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-9q^{-46}+11q^{-48}-8q^{-50}+3q^{-52}+5q^{-54}-13q^{-56}+21q^{-58}-19q^{-60}+12q^{-62}-2q^{-64}-10q^{-66}+18q^{-68}-17q^{-70}+14q^{-72}-2q^{-74}-7q^{-76}+13q^{-78}-9q^{-80}-q^{-82}+12q^{-84}-19q^{-86}+18q^{-88}-9q^{-90}-4q^{-92}+21q^{-94}-28q^{-96}+31q^{-98}-20q^{-100}+6q^{-102}+11q^{-104}-21q^{-106}+26q^{-108}-18q^{-110}+12q^{-112}+2q^{-114}-10q^{-116}+15q^{-118}-10q^{-120}-2q^{-122}+9q^{-124}-16q^{-126}+10q^{-128}-3q^{-130}-12q^{-132}+18q^{-134}-20q^{-136}+15q^{-138}-10q^{-140}-9q^{-142}+13q^{-144}-16q^{-146}+13q^{-148}-9q^{-150}+2q^{-152}+3q^{-154}-4q^{-156}+6q^{-158}-6q^{-160}+4q^{-162}-q^{-164}+q^{-168}-2q^{-170}+2q^{-172}+q^{-176}$

.

Computer Talk

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["9 10"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-8t+9-8t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+8z^{2}+1$

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]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

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["9 10"];

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]=

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

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]=

{}

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:

(8, 22)

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 $32$ $176$ $512$ ${\frac {3856}{3}}$ ${\frac {656}{3}}$ $5632$ ${\frac {30752}{3}}$ ${\frac {5504}{3}}$ $1520$ ${\frac {16384}{3}}$ $15488$ ${\frac {123392}{3}}$ ${\frac {20992}{3}}$ ${\frac {1241164}{15}}$ ${\frac {6064}{15}}$ ${\frac {1589776}{45}}$ ${\frac {5108}{9}}$ ${\frac {72844}{15}}$

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=$4 is the signature of 9 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{31}-q^{30}+3q^{28}-4q^{27}-2q^{26}+10q^{25}-10q^{24}-7q^{23}+22q^{22}-14q^{21}-15q^{20}+32q^{19}-13q^{18}-22q^{17}+35q^{16}-11q^{15}-22q^{14}+28q^{13}-5q^{12}-16q^{11}+15q^{10}-8q^{8}+5q^{7}+q^{6}-2q^{5}+q^{4}$
3	$-q^{60}+q^{59}-2q^{56}+3q^{55}-q^{53}-5q^{52}+8q^{51}+5q^{50}-7q^{49}-16q^{48}+16q^{47}+21q^{46}-13q^{45}-37q^{44}+13q^{43}+50q^{42}-10q^{41}-62q^{40}-q^{39}+76q^{38}+7q^{37}-81q^{36}-22q^{35}+94q^{34}+24q^{33}-90q^{32}-37q^{31}+94q^{30}+36q^{29}-84q^{28}-43q^{27}+77q^{26}+41q^{25}-60q^{24}-41q^{23}+46q^{22}+35q^{21}-28q^{20}-32q^{19}+19q^{18}+21q^{17}-6q^{16}-17q^{15}+4q^{14}+9q^{13}-6q^{11}+q^{10}+2q^{9}+q^{8}-2q^{7}+q^{6}$
4	$q^{98}-q^{97}-q^{94}+3q^{93}-3q^{92}+q^{91}+2q^{90}-5q^{89}+6q^{88}-8q^{87}+2q^{86}+9q^{85}-6q^{84}+11q^{83}-25q^{82}-5q^{81}+21q^{80}+7q^{79}+34q^{78}-55q^{77}-35q^{76}+20q^{75}+28q^{74}+97q^{73}-72q^{72}-83q^{71}-22q^{70}+27q^{69}+193q^{68}-49q^{67}-122q^{66}-94q^{65}-14q^{64}+284q^{63}+8q^{62}-125q^{61}-172q^{60}-79q^{59}+349q^{58}+69q^{57}-107q^{56}-232q^{55}-137q^{54}+379q^{53}+114q^{52}-80q^{51}-261q^{50}-180q^{49}+373q^{48}+138q^{47}-44q^{46}-252q^{45}-204q^{44}+318q^{43}+139q^{42}+5q^{41}-203q^{40}-203q^{39}+220q^{38}+108q^{37}+50q^{36}-120q^{35}-168q^{34}+113q^{33}+55q^{32}+66q^{31}-43q^{30}-108q^{29}+44q^{28}+8q^{27}+48q^{26}-2q^{25}-51q^{24}+16q^{23}-10q^{22}+23q^{21}+5q^{20}-20q^{19}+8q^{18}-7q^{17}+8q^{16}+3q^{15}-7q^{14}+3q^{13}-2q^{12}+2q^{11}+q^{10}-2q^{9}+q^{8}$

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.

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