10 61

10_60

10_62

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 61's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 61 at Knotilus!
	10_61 is also known as the pretzel knot P(4,3,3).

Knot presentations

Planar diagram presentation	X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X20,14,1,13 X12,20,13,19
Gauss code	1, -3, 2, -8, 6, -7, 5, -1, 3, -2, 4, -10, 9, -5, 7, -6, 8, -4, 10, -9
Dowker-Thistlethwaite code	8 10 16 14 2 18 20 6 4 12
Conway Notation	[4,3,3]

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

[edit Notes on presentations of 10 61]

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["10 61"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X20,14,1,13 X12,20,13,19

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

8 10 16 14 2 18 20 6 4 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[4,3,3]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-1][-11] Hyperbolic Volume 8.45858 A-Polynomial See Data:10 61/A-polynomial

[edit Notes for 10 61's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 10 61's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+5t^{2}-6t+7-6t^{-1}+5t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-7z^{4}-4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$q^{8}-2q^{7}+3q^{6}-4q^{5}+5q^{4}-5q^{3}+4q^{2}-4q+3-q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}a^{-4}-5z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+z^{4}-8z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+4z^{2}-5a^{-2}+a^{-4}+a^{-6}+4$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+3z^{8}a^{-4}+z^{8}-5z^{7}a^{-1}+5z^{7}a^{-5}-22z^{6}a^{-2}-10z^{6}a^{-4}+5z^{6}a^{-6}-7z^{6}+6z^{5}a^{-1}-16z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}+38z^{4}a^{-2}+5z^{4}a^{-4}-13z^{4}a^{-6}+3z^{4}a^{-8}+17z^{4}+z^{3}a^{-1}+26z^{3}a^{-3}+17z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-24z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}-16z^{2}-2za^{-1}-8za^{-3}-6za^{-5}+5a^{-2}+a^{-4}-a^{-6}+4$
The A2 invariant	$q^{6}+q^{4}+2q^{2}+2-q^{-4}-3q^{-6}-2q^{-8}+2q^{-14}+q^{-24}$
The G2 invariant	$q^{26}+3q^{22}-3q^{20}+3q^{18}-q^{16}+7q^{12}-9q^{10}+10q^{8}-3q^{6}+q^{4}+8q^{2}-12+11q^{-2}-2q^{-4}+5q^{-8}-9q^{-10}+4q^{-12}+5q^{-14}-4q^{-16}+q^{-18}-5q^{-20}-q^{-22}+4q^{-24}-8q^{-26}+4q^{-28}-9q^{-30}+5q^{-32}+q^{-34}-7q^{-36}+6q^{-38}-12q^{-40}+8q^{-42}-5q^{-44}-3q^{-46}+7q^{-48}-9q^{-50}+5q^{-52}+3q^{-54}-3q^{-56}+4q^{-58}-q^{-60}-4q^{-62}+6q^{-64}-3q^{-66}+3q^{-68}+q^{-70}-2q^{-72}+6q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}-q^{-84}+q^{-86}+q^{-88}-2q^{-90}+4q^{-92}-3q^{-94}+2q^{-96}-q^{-98}-q^{-100}-3q^{-104}+3q^{-106}-q^{-108}+q^{-110}-q^{-114}+2q^{-116}-2q^{-118}+2q^{-120}-q^{-122}-q^{-128}+q^{-130}-q^{-132}+q^{-134}$

Further Quantum Invariants

Further quantum knot invariants for 10_61.

A1 Invariants.

Weight	Invariant
1	$q^{5}+2q-q^{-1}-q^{-5}+q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}$
2	$q^{18}-q^{14}+2q^{12}+2q^{10}-3q^{8}+2q^{4}-3q^{2}-2+2q^{-2}-q^{-6}+3q^{-8}+3q^{-10}-q^{-12}-q^{-14}+2q^{-16}-q^{-18}-3q^{-20}+2q^{-22}+q^{-24}-q^{-26}+q^{-30}-q^{-32}-q^{-34}+q^{-36}-q^{-38}+q^{-40}-q^{-44}+q^{-46}$
3	$q^{39}-q^{35}-q^{33}+2q^{31}+3q^{29}-5q^{25}-2q^{23}+4q^{21}+5q^{19}-3q^{17}-7q^{15}-3q^{13}+5q^{11}+5q^{9}-2q^{7}-5q^{5}-q^{3}+7q+6q^{-1}-q^{-3}-5q^{-5}+q^{-7}+5q^{-9}+2q^{-11}-6q^{-13}-3q^{-15}+3q^{-17}+3q^{-19}-4q^{-21}-4q^{-23}+5q^{-25}+6q^{-27}-5q^{-29}-8q^{-31}+3q^{-33}+8q^{-35}-8q^{-39}-4q^{-41}+3q^{-43}+7q^{-45}+3q^{-47}-6q^{-49}-6q^{-51}+6q^{-53}+9q^{-55}-3q^{-57}-8q^{-59}+5q^{-63}-3q^{-67}+q^{-69}+q^{-71}-2q^{-75}+q^{-79}-q^{-85}+q^{-87}$
4	$q^{68}-q^{64}-q^{62}-q^{60}+3q^{58}+3q^{56}+q^{54}-2q^{52}-8q^{50}-2q^{48}+4q^{46}+9q^{44}+7q^{42}-7q^{40}-11q^{38}-11q^{36}+2q^{34}+16q^{32}+11q^{30}+2q^{28}-16q^{26}-18q^{24}-q^{22}+13q^{20}+25q^{18}+9q^{16}-14q^{14}-21q^{12}-14q^{10}+15q^{8}+28q^{6}+12q^{4}-11q^{2}-31-17q^{-2}+13q^{-4}+26q^{-6}+17q^{-8}-15q^{-10}-29q^{-12}-12q^{-14}+14q^{-16}+29q^{-18}+12q^{-20}-17q^{-22}-22q^{-24}-5q^{-26}+21q^{-28}+17q^{-30}-7q^{-32}-17q^{-34}-9q^{-36}+13q^{-38}+13q^{-40}-9q^{-42}-16q^{-44}-2q^{-46}+22q^{-48}+20q^{-50}-14q^{-52}-28q^{-54}-12q^{-56}+21q^{-58}+34q^{-60}+6q^{-62}-25q^{-64}-32q^{-66}-10q^{-68}+25q^{-70}+33q^{-72}+11q^{-74}-21q^{-76}-40q^{-78}-12q^{-80}+26q^{-82}+37q^{-84}+11q^{-86}-33q^{-88}-32q^{-90}+2q^{-92}+29q^{-94}+22q^{-96}-14q^{-98}-21q^{-100}-4q^{-102}+12q^{-104}+14q^{-106}-4q^{-108}-7q^{-110}-3q^{-112}+2q^{-114}+4q^{-116}-4q^{-118}+q^{-120}+q^{-122}-3q^{-128}+2q^{-130}-q^{-138}+q^{-140}$
5	$q^{105}-q^{101}-q^{99}-q^{97}+3q^{93}+4q^{91}+q^{89}-2q^{87}-5q^{85}-8q^{83}-3q^{81}+6q^{79}+12q^{77}+11q^{75}+3q^{73}-11q^{71}-20q^{69}-17q^{67}-2q^{65}+17q^{63}+28q^{61}+22q^{59}+q^{57}-25q^{55}-39q^{53}-27q^{51}+3q^{49}+34q^{47}+49q^{45}+35q^{43}-7q^{41}-46q^{39}-57q^{37}-33q^{35}+14q^{33}+61q^{31}+70q^{29}+30q^{27}-31q^{25}-74q^{23}-75q^{21}-23q^{19}+49q^{17}+88q^{15}+67q^{13}+3q^{11}-70q^{9}-99q^{7}-58q^{5}+24q^{3}+92q+97q^{-1}+35q^{-3}-55q^{-5}-106q^{-7}-81q^{-9}+5q^{-11}+91q^{-13}+114q^{-15}+49q^{-17}-51q^{-19}-116q^{-21}-91q^{-23}+5q^{-25}+99q^{-27}+112q^{-29}+38q^{-31}-66q^{-33}-116q^{-35}-73q^{-37}+30q^{-39}+104q^{-41}+88q^{-43}+5q^{-45}-78q^{-47}-90q^{-49}-24q^{-51}+51q^{-53}+71q^{-55}+28q^{-57}-33q^{-59}-48q^{-61}-16q^{-63}+29q^{-65}+31q^{-67}-12q^{-69}-45q^{-71}-26q^{-73}+29q^{-75}+73q^{-77}+49q^{-79}-33q^{-81}-99q^{-83}-81q^{-85}+9q^{-87}+99q^{-89}+117q^{-91}+43q^{-93}-66q^{-95}-127q^{-97}-94q^{-99}+2q^{-101}+96q^{-103}+127q^{-105}+74q^{-107}-37q^{-109}-129q^{-111}-128q^{-113}-35q^{-115}+93q^{-117}+155q^{-119}+94q^{-121}-42q^{-123}-145q^{-125}-124q^{-127}-2q^{-129}+110q^{-131}+125q^{-133}+32q^{-135}-78q^{-137}-108q^{-139}-41q^{-141}+51q^{-143}+82q^{-145}+38q^{-147}-32q^{-149}-61q^{-151}-28q^{-153}+24q^{-155}+39q^{-157}+16q^{-159}-15q^{-161}-25q^{-163}-9q^{-165}+12q^{-167}+15q^{-169}+3q^{-171}-6q^{-173}-5q^{-175}-q^{-177}+q^{-179}+2q^{-181}-q^{-185}+q^{-187}-q^{-191}-q^{-193}+q^{-195}-q^{-203}+q^{-205}$

A2 Invariants.

Weight	Invariant
1,0	$q^{6}+q^{4}+2q^{2}+2-q^{-4}-3q^{-6}-2q^{-8}+2q^{-14}+q^{-24}$
1,1	$q^{20}+6q^{16}-6q^{14}+14q^{12}-18q^{10}+26q^{8}-24q^{6}+22q^{4}-22q^{2}+6-15q^{-4}+16q^{-6}-26q^{-8}+34q^{-10}-32q^{-12}+34q^{-14}-28q^{-16}+28q^{-18}-13q^{-20}+12q^{-22}+2q^{-24}-4q^{-26}+6q^{-28}-8q^{-30}-6q^{-34}+3q^{-36}-2q^{-38}+4q^{-40}-8q^{-42}+7q^{-44}-6q^{-46}+6q^{-48}-6q^{-50}+5q^{-52}-2q^{-54}+4q^{-56}-4q^{-58}+3q^{-60}-2q^{-62}+2q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{20}+q^{18}+q^{16}+q^{14}+3q^{12}+3q^{10}+2q^{8}-q^{6}-q^{4}-3q^{2}-5-7q^{-2}-6q^{-4}-2q^{-6}+3q^{-10}+7q^{-12}+9q^{-14}+6q^{-16}+4q^{-18}-2q^{-22}-4q^{-24}-3q^{-26}-2q^{-28}-2q^{-30}+2q^{-32}+4q^{-34}+q^{-36}-2q^{-38}-q^{-40}-2q^{-42}-2q^{-44}-q^{-46}+q^{-48}+2q^{-50}+q^{-60}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{12}+3q^{8}+2q^{6}+2q^{4}+3q^{2}+1-q^{-2}-2q^{-6}-4q^{-8}-2q^{-10}-4q^{-12}-2q^{-14}+2q^{-20}+4q^{-22}+3q^{-24}+q^{-26}-q^{-32}-2q^{-34}+q^{-36}+q^{-38}-q^{-40}+q^{-42}+q^{-44}-2q^{-46}+2q^{-50}-q^{-52}-q^{-54}+q^{-56}$
1,0,0	$q^{7}+q^{5}+3q^{3}+2q+3q^{-1}-q^{-5}-4q^{-7}-3q^{-9}-3q^{-11}-q^{-13}+q^{-15}+q^{-17}+2q^{-19}+q^{-23}-q^{-25}+q^{-27}+q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{14}+q^{12}+3q^{10}+5q^{8}+6q^{6}+6q^{4}+7q^{2}+2-q^{-2}-5q^{-4}-9q^{-6}-12q^{-8}-10q^{-10}-6q^{-12}-5q^{-14}+6q^{-18}+8q^{-20}+2q^{-22}+6q^{-24}+6q^{-26}+q^{-28}+q^{-30}+2q^{-32}+q^{-34}-2q^{-36}-q^{-38}-2q^{-40}-3q^{-42}-3q^{-44}+q^{-46}+q^{-48}-q^{-50}+2q^{-52}+2q^{-54}-q^{-58}+q^{-62}-q^{-66}+q^{-70}$
1,0,0,0	$q^{8}+q^{6}+3q^{4}+3q^{2}+3+3q^{-2}-q^{-6}-4q^{-8}-4q^{-10}-4q^{-12}-3q^{-14}-q^{-16}+2q^{-20}+q^{-22}+2q^{-24}+q^{-28}+q^{-34}+q^{-38}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{14}+3q^{10}+6q^{6}+6q^{2}+6q^{-2}-2q^{-4}+2q^{-6}-4q^{-8}-2q^{-10}-5q^{-12}-7q^{-14}-3q^{-16}-7q^{-18}+q^{-20}-7q^{-22}+6q^{-24}-3q^{-26}+10q^{-28}-2q^{-30}+8q^{-32}-2q^{-34}+5q^{-36}-2q^{-38}+q^{-40}-2q^{-42}-q^{-44}+q^{-46}-q^{-48}+2q^{-50}-2q^{-52}+3q^{-54}-2q^{-56}+2q^{-58}-2q^{-60}+2q^{-62}-2q^{-64}+q^{-66}-q^{-68}+2q^{-70}-q^{-72}-q^{-76}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{26}+3q^{22}-3q^{20}+3q^{18}-q^{16}+7q^{12}-9q^{10}+10q^{8}-3q^{6}+q^{4}+8q^{2}-12+11q^{-2}-2q^{-4}+5q^{-8}-9q^{-10}+4q^{-12}+5q^{-14}-4q^{-16}+q^{-18}-5q^{-20}-q^{-22}+4q^{-24}-8q^{-26}+4q^{-28}-9q^{-30}+5q^{-32}+q^{-34}-7q^{-36}+6q^{-38}-12q^{-40}+8q^{-42}-5q^{-44}-3q^{-46}+7q^{-48}-9q^{-50}+5q^{-52}+3q^{-54}-3q^{-56}+4q^{-58}-q^{-60}-4q^{-62}+6q^{-64}-3q^{-66}+3q^{-68}+q^{-70}-2q^{-72}+6q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}-q^{-84}+q^{-86}+q^{-88}-2q^{-90}+4q^{-92}-3q^{-94}+2q^{-96}-q^{-98}-q^{-100}-3q^{-104}+3q^{-106}-q^{-108}+q^{-110}-q^{-114}+2q^{-116}-2q^{-118}+2q^{-120}-q^{-122}-q^{-128}+q^{-130}-q^{-132}+q^{-134}$

.

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["10 61"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+5t^{2}-6t+7-6t^{-1}+5t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-7z^{4}-4z^{2}+1$

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{8}-2q^{7}+3q^{6}-4q^{5}+5q^{4}-5q^{3}+4q^{2}-4q+3-q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-z^{6}a^{-4}-5z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+z^{4}-8z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+4z^{2}-5a^{-2}+a^{-4}+a^{-6}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+3z^{8}a^{-4}+z^{8}-5z^{7}a^{-1}+5z^{7}a^{-5}-22z^{6}a^{-2}-10z^{6}a^{-4}+5z^{6}a^{-6}-7z^{6}+6z^{5}a^{-1}-16z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}+38z^{4}a^{-2}+5z^{4}a^{-4}-13z^{4}a^{-6}+3z^{4}a^{-8}+17z^{4}+z^{3}a^{-1}+26z^{3}a^{-3}+17z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-24z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}-16z^{2}-2za^{-1}-8za^{-3}-6za^{-5}+5a^{-2}+a^{-4}-a^{-6}+4$

"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["10 61"];

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]=

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

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:

(-4, -5)

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 $-16$ $-40$ $128$ ${\frac {568}{3}}$ ${\frac {368}{3}}$ $640$ ${\frac {3536}{3}}$ ${\frac {1088}{3}}$ $248$ $-{\frac {2048}{3}}$ $800$ $-{\frac {9088}{3}}$ $-{\frac {5888}{3}}$ $-{\frac {8822}{15}}$ ${\frac {28208}{15}}$ $-{\frac {156128}{45}}$ ${\frac {5894}{9}}$ $-{\frac {16022}{15}}$

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 10 61. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{22}-2q^{21}+q^{20}+2q^{19}-4q^{18}+3q^{17}-4q^{15}+5q^{14}-q^{13}-5q^{12}+7q^{11}-10q^{9}+9q^{8}+3q^{7}-13q^{6}+9q^{5}+7q^{4}-13q^{3}+5q^{2}+8q-11+q^{-1}+7q^{-2}-6q^{-3}-q^{-4}+4q^{-5}-q^{-6}-q^{-7}+q^{-8}$
3	$q^{42}-2q^{41}+q^{40}+2q^{38}-3q^{37}-q^{36}+2q^{35}+3q^{34}-3q^{33}-5q^{32}+5q^{31}+8q^{30}-8q^{29}-13q^{28}+10q^{27}+20q^{26}-11q^{25}-25q^{24}+10q^{23}+29q^{22}-7q^{21}-29q^{20}+3q^{19}+25q^{18}+q^{17}-21q^{16}-2q^{15}+14q^{14}+4q^{13}-10q^{12}-3q^{11}+5q^{10}+4q^{9}-3q^{8}-3q^{7}-q^{6}+q^{5}+5q^{4}-5q^{2}-5q+9+7q^{-1}-4q^{-2}-13q^{-3}+5q^{-4}+10q^{-5}+3q^{-6}-13q^{-7}-3q^{-8}+6q^{-9}+7q^{-10}-5q^{-11}-4q^{-12}+4q^{-14}-q^{-16}-q^{-17}+q^{-18}$
4	$q^{68}-2q^{67}+q^{66}+3q^{63}-7q^{62}+4q^{61}+q^{59}+3q^{58}-12q^{57}+12q^{56}-2q^{55}-4q^{54}-q^{53}-9q^{52}+30q^{51}-4q^{50}-20q^{49}-18q^{48}-2q^{47}+66q^{46}+3q^{45}-47q^{44}-52q^{43}-3q^{42}+110q^{41}+29q^{40}-58q^{39}-90q^{38}-31q^{37}+129q^{36}+61q^{35}-36q^{34}-98q^{33}-66q^{32}+107q^{31}+68q^{30}-5q^{29}-70q^{28}-79q^{27}+74q^{26}+52q^{25}+9q^{24}-36q^{23}-77q^{22}+50q^{21}+38q^{20}+16q^{19}-14q^{18}-77q^{17}+28q^{16}+30q^{15}+26q^{14}+10q^{13}-73q^{12}+2q^{11}+13q^{10}+31q^{9}+39q^{8}-56q^{7}-13q^{6}-13q^{5}+14q^{4}+53q^{3}-24q^{2}-4q-26-16q^{-1}+39q^{-2}-4q^{-3}+19q^{-4}-10q^{-5}-29q^{-6}+10q^{-7}-11q^{-8}+26q^{-9}+13q^{-10}-13q^{-11}-2q^{-12}-25q^{-13}+9q^{-14}+15q^{-15}+5q^{-16}+7q^{-17}-20q^{-18}-5q^{-19}+2q^{-20}+5q^{-21}+11q^{-22}-6q^{-23}-3q^{-24}-3q^{-25}-q^{-26}+5q^{-27}-q^{-30}-q^{-31}+q^{-32}$
5	$q^{100}-2q^{99}+q^{98}+q^{95}-q^{94}-2q^{93}+2q^{92}+q^{91}-2q^{90}+2q^{89}+q^{88}-3q^{87}-3q^{85}-3q^{84}+11q^{83}+13q^{82}-6q^{81}-21q^{80}-19q^{79}+7q^{78}+42q^{77}+36q^{76}-21q^{75}-73q^{74}-52q^{73}+36q^{72}+112q^{71}+80q^{70}-52q^{69}-165q^{68}-119q^{67}+66q^{66}+222q^{65}+173q^{64}-67q^{63}-277q^{62}-241q^{61}+45q^{60}+325q^{59}+309q^{58}-6q^{57}-339q^{56}-369q^{55}-48q^{54}+324q^{53}+401q^{52}+105q^{51}-286q^{50}-400q^{49}-142q^{48}+228q^{47}+368q^{46}+166q^{45}-177q^{44}-326q^{43}-160q^{42}+138q^{41}+278q^{40}+148q^{39}-111q^{38}-244q^{37}-136q^{36}+94q^{35}+223q^{34}+129q^{33}-78q^{32}-201q^{31}-138q^{30}+49q^{29}+191q^{28}+144q^{27}-17q^{26}-158q^{25}-158q^{24}-26q^{23}+125q^{22}+156q^{21}+66q^{20}-75q^{19}-142q^{18}-100q^{17}+22q^{16}+113q^{15}+116q^{14}+29q^{13}-68q^{12}-113q^{11}-72q^{10}+17q^{9}+91q^{8}+94q^{7}+32q^{6}-48q^{5}-95q^{4}-69q^{3}+5q^{2}+69q+83+42q^{-1}-33q^{-2}-74q^{-3}-63q^{-4}-13q^{-5}+42q^{-6}+71q^{-7}+40q^{-8}-10q^{-9}-42q^{-10}-52q^{-11}-30q^{-12}+19q^{-13}+41q^{-14}+33q^{-15}+19q^{-16}-12q^{-17}-39q^{-18}-28q^{-19}-6q^{-20}+9q^{-21}+30q^{-22}+27q^{-23}+3q^{-24}-14q^{-25}-21q^{-26}-22q^{-27}+15q^{-29}+17q^{-30}+12q^{-31}-16q^{-33}-11q^{-34}-4q^{-35}+2q^{-36}+9q^{-37}+9q^{-38}-2q^{-39}-3q^{-40}-3q^{-41}-4q^{-42}+4q^{-44}+q^{-45}-q^{-48}-q^{-49}+q^{-50}$
6	$q^{138}-2q^{137}+q^{136}+q^{133}-3q^{132}+4q^{131}-4q^{130}+3q^{129}-2q^{128}+6q^{126}-8q^{125}+5q^{124}-8q^{123}+5q^{122}-2q^{121}+7q^{120}+16q^{119}-22q^{118}-6q^{117}-14q^{116}+15q^{115}+11q^{114}+25q^{113}+17q^{112}-64q^{111}-35q^{110}-5q^{109}+64q^{108}+55q^{107}+35q^{106}-31q^{105}-165q^{104}-76q^{103}+60q^{102}+197q^{101}+151q^{100}+10q^{99}-178q^{98}-368q^{97}-143q^{96}+203q^{95}+463q^{94}+355q^{93}-9q^{92}-428q^{91}-728q^{90}-331q^{89}+330q^{88}+840q^{87}+739q^{86}+135q^{85}-629q^{84}-1192q^{83}-730q^{82}+232q^{81}+1114q^{80}+1205q^{79}+526q^{78}-537q^{77}-1478q^{76}-1176q^{75}-139q^{74}+1024q^{73}+1417q^{72}+922q^{71}-169q^{70}-1348q^{69}-1309q^{68}-475q^{67}+672q^{66}+1211q^{65}+990q^{64}+126q^{63}-996q^{62}-1084q^{61}-516q^{60}+420q^{59}+881q^{58}+801q^{57}+171q^{56}-769q^{55}-841q^{54}-417q^{53}+355q^{52}+718q^{51}+681q^{50}+170q^{49}-677q^{48}-773q^{47}-444q^{46}+266q^{45}+649q^{44}+716q^{43}+307q^{42}-516q^{41}-740q^{40}-589q^{39}+32q^{38}+483q^{37}+740q^{36}+520q^{35}-220q^{34}-578q^{33}-681q^{32}-258q^{31}+184q^{30}+612q^{29}+653q^{28}+119q^{27}-268q^{26}-601q^{25}-455q^{24}-162q^{23}+315q^{22}+594q^{21}+363q^{20}+100q^{19}-328q^{18}-438q^{17}-406q^{16}-59q^{15}+318q^{14}+373q^{13}+355q^{12}+33q^{11}-180q^{10}-389q^{9}-311q^{8}-45q^{7}+126q^{6}+326q^{5}+247q^{4}+151q^{3}-111q^{2}-260q-231-167q^{-1}+57q^{-2}+150q^{-3}+265q^{-4}+165q^{-5}+3q^{-6}-104q^{-7}-216q^{-8}-153q^{-9}-104q^{-10}+87q^{-11}+169q^{-12}+150q^{-13}+115q^{-14}-25q^{-15}-87q^{-16}-184q^{-17}-102q^{-18}-16q^{-19}+43q^{-20}+129q^{-21}+102q^{-22}+83q^{-23}-48q^{-24}-68q^{-25}-84q^{-26}-89q^{-27}-8q^{-28}+29q^{-29}+95q^{-30}+44q^{-31}+47q^{-32}+4q^{-33}-56q^{-34}-52q^{-35}-53q^{-36}+4q^{-37}-5q^{-38}+48q^{-39}+49q^{-40}+19q^{-41}+2q^{-42}-26q^{-43}-17q^{-44}-45q^{-45}-4q^{-46}+12q^{-47}+18q^{-48}+20q^{-49}+11q^{-50}+11q^{-51}-23q^{-52}-11q^{-53}-9q^{-54}-3q^{-55}+2q^{-56}+7q^{-57}+14q^{-58}-3q^{-59}-3q^{-61}-3q^{-62}-4q^{-63}-q^{-64}+5q^{-65}+q^{-67}-q^{-70}-q^{-71}+q^{-72}$

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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