10 111

10_110

10_112

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10 111 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X14,8,15,7 X8,14,9,13 X2,10,3,9 X18,12,19,11 X16,5,17,6 X4,17,5,18 X20,16,1,15 X12,20,13,19
Gauss code	1, -5, 2, -8, 7, -1, 3, -4, 5, -2, 6, -10, 4, -3, 9, -7, 8, -6, 10, -9
Dowker-Thistlethwaite code	6 10 16 14 2 18 8 20 4 12
Conway Notation	[2.2.20.2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [1][-13] Hyperbolic Volume 14.265 A-Polynomial See Data:10 111/A-polynomial

[edit Notes for 10 111'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 10 111's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-17t+21-17t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-10q^{7}+12q^{6}-13q^{5}+12q^{4}-9q^{3}+7q^{2}-3q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-2z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-2}+z^{2}a^{-4}-4z^{2}a^{-6}+2z^{2}a^{-8}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+10z^{8}a^{-6}+6z^{8}a^{-8}+3z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+7z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-4}-26z^{6}a^{-6}-11z^{6}a^{-8}+5z^{6}a^{-10}-8z^{5}a^{-3}-20z^{5}a^{-5}-28z^{5}a^{-7}-13z^{5}a^{-9}+3z^{5}a^{-11}-3z^{4}a^{-2}+3z^{4}a^{-4}+22z^{4}a^{-6}+10z^{4}a^{-8}-5z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-3}+19z^{3}a^{-5}+30z^{3}a^{-7}+13z^{3}a^{-9}-3z^{3}a^{-11}+3z^{2}a^{-2}-2z^{2}a^{-4}-10z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-za^{-3}-7za^{-5}-10za^{-7}-4za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$
The A2 invariant	$1-q^{-2}+q^{-4}+2q^{-6}-q^{-8}+4q^{-10}-q^{-12}+q^{-14}-3q^{-18}+q^{-20}-3q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}-2q^{-4}+6q^{-6}-10q^{-8}+13q^{-10}-11q^{-12}+q^{-14}+20q^{-16}-43q^{-18}+67q^{-20}-74q^{-22}+48q^{-24}+8q^{-26}-84q^{-28}+152q^{-30}-169q^{-32}+131q^{-34}-35q^{-36}-85q^{-38}+181q^{-40}-207q^{-42}+155q^{-44}-36q^{-46}-85q^{-48}+163q^{-50}-153q^{-52}+71q^{-54}+49q^{-56}-142q^{-58}+171q^{-60}-123q^{-62}+10q^{-64}+117q^{-66}-214q^{-68}+239q^{-70}-182q^{-72}+57q^{-74}+88q^{-76}-213q^{-78}+259q^{-80}-226q^{-82}+114q^{-84}+29q^{-86}-152q^{-88}+201q^{-90}-163q^{-92}+57q^{-94}+65q^{-96}-143q^{-98}+139q^{-100}-64q^{-102}-48q^{-104}+143q^{-106}-173q^{-108}+137q^{-110}-44q^{-112}-60q^{-114}+134q^{-116}-157q^{-118}+131q^{-120}-68q^{-122}+2q^{-124}+50q^{-126}-77q^{-128}+77q^{-130}-60q^{-132}+38q^{-134}-12q^{-136}-9q^{-138}+21q^{-140}-28q^{-142}+24q^{-144}-17q^{-146}+10q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants

Further quantum knot invariants for 10_111.

A1 Invariants.

Weight	Invariant
1	$q-2q^{-1}+4q^{-3}-2q^{-5}+3q^{-7}-q^{-9}-q^{-11}+2q^{-13}-4q^{-15}+3q^{-17}-2q^{-19}+q^{-21}$
2	$q^{6}-2q^{4}-q^{2}+9-5q^{-2}-13q^{-4}+20q^{-6}+5q^{-8}-27q^{-10}+15q^{-12}+20q^{-14}-27q^{-16}+23q^{-20}-13q^{-22}-12q^{-24}+14q^{-26}+7q^{-28}-19q^{-30}-4q^{-32}+27q^{-34}-14q^{-36}-20q^{-38}+29q^{-40}-2q^{-42}-20q^{-44}+14q^{-46}+3q^{-48}-8q^{-50}+4q^{-52}-2q^{-56}+q^{-58}$
3	$q^{15}-2q^{13}-q^{11}+4q^{9}+6q^{7}-8q^{5}-19q^{3}+10q+41q^{-1}+5q^{-3}-63q^{-5}-44q^{-7}+77q^{-9}+92q^{-11}-52q^{-13}-143q^{-15}+3q^{-17}+172q^{-19}+66q^{-21}-166q^{-23}-129q^{-25}+129q^{-27}+177q^{-29}-79q^{-31}-197q^{-33}+29q^{-35}+190q^{-37}+20q^{-39}-176q^{-41}-54q^{-43}+143q^{-45}+87q^{-47}-112q^{-49}-116q^{-51}+62q^{-53}+146q^{-55}-3q^{-57}-165q^{-59}-65q^{-61}+166q^{-63}+132q^{-65}-132q^{-67}-182q^{-69}+82q^{-71}+196q^{-73}-25q^{-75}-170q^{-77}-27q^{-79}+125q^{-81}+46q^{-83}-71q^{-85}-43q^{-87}+31q^{-89}+27q^{-91}-11q^{-93}-13q^{-95}+5q^{-97}+3q^{-99}-q^{-101}+q^{-105}-2q^{-109}+q^{-111}$
4	$q^{28}-2q^{26}-q^{24}+4q^{22}+q^{20}+3q^{18}-14q^{16}-13q^{14}+18q^{12}+28q^{10}+39q^{8}-48q^{6}-105q^{4}-36q^{2}+77+235q^{-2}+81q^{-4}-207q^{-6}-342q^{-8}-196q^{-10}+385q^{-12}+565q^{-14}+197q^{-16}-481q^{-18}-899q^{-20}-186q^{-22}+729q^{-24}+1079q^{-26}+279q^{-28}-1115q^{-30}-1227q^{-32}-137q^{-34}+1339q^{-36}+1440q^{-38}-238q^{-40}-1562q^{-42}-1334q^{-44}+558q^{-46}+1841q^{-48}+888q^{-50}-940q^{-52}-1780q^{-54}-397q^{-56}+1376q^{-58}+1351q^{-60}-187q^{-62}-1488q^{-64}-820q^{-66}+761q^{-68}+1266q^{-70}+225q^{-72}-1053q^{-74}-933q^{-76}+282q^{-78}+1145q^{-80}+592q^{-82}-625q^{-84}-1151q^{-86}-392q^{-88}+964q^{-90}+1206q^{-92}+173q^{-94}-1266q^{-96}-1391q^{-98}+251q^{-100}+1594q^{-102}+1336q^{-104}-641q^{-106}-1978q^{-108}-916q^{-110}+1025q^{-112}+1946q^{-114}+507q^{-116}-1405q^{-118}-1474q^{-120}-104q^{-122}+1345q^{-124}+1034q^{-126}-301q^{-128}-949q^{-130}-607q^{-132}+372q^{-134}+637q^{-136}+192q^{-138}-230q^{-140}-367q^{-142}-30q^{-144}+166q^{-146}+118q^{-148}+17q^{-150}-101q^{-152}-31q^{-154}+17q^{-156}+19q^{-158}+22q^{-160}-20q^{-162}-5q^{-164}+q^{-166}-q^{-168}+7q^{-170}-3q^{-172}+q^{-174}-2q^{-178}+q^{-180}$

A2 Invariants.

Weight	Invariant
1,0	$1-q^{-2}+q^{-4}+2q^{-6}-q^{-8}+4q^{-10}-q^{-12}+q^{-14}-3q^{-18}+q^{-20}-3q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
1,1	$q^{4}-4q^{2}+14-36q^{-2}+80q^{-4}-146q^{-6}+254q^{-8}-392q^{-10}+548q^{-12}-682q^{-14}+786q^{-16}-808q^{-18}+720q^{-20}-514q^{-22}+216q^{-24}+150q^{-26}-543q^{-28}+906q^{-30}-1206q^{-32}+1396q^{-34}-1471q^{-36}+1408q^{-38}-1222q^{-40}+940q^{-42}-589q^{-44}+216q^{-46}+126q^{-48}-396q^{-50}+574q^{-52}-656q^{-54}+650q^{-56}-578q^{-58}+486q^{-60}-388q^{-62}+292q^{-64}-216q^{-66}+161q^{-68}-118q^{-70}+82q^{-72}-54q^{-74}+34q^{-76}-20q^{-78}+10q^{-80}-4q^{-82}+q^{-84}$
2,0	$q^{4}-q^{2}-2+3q^{-2}+5q^{-4}-q^{-6}-7q^{-8}+2q^{-10}+13q^{-12}-3q^{-14}-10q^{-16}+6q^{-18}+11q^{-20}-5q^{-22}-8q^{-24}+7q^{-26}+3q^{-28}-7q^{-30}+q^{-32}+5q^{-34}-7q^{-36}-4q^{-38}+9q^{-40}-7q^{-42}-10q^{-44}+6q^{-46}+11q^{-48}-6q^{-50}-9q^{-52}+11q^{-54}+6q^{-56}-6q^{-58}-5q^{-60}+5q^{-62}+2q^{-64}-2q^{-66}-q^{-68}-q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-2q^{-2}+2q^{-4}+4q^{-6}-8q^{-8}+7q^{-10}+7q^{-12}-15q^{-14}+17q^{-16}+8q^{-18}-17q^{-20}+16q^{-22}+7q^{-24}-20q^{-26}+q^{-28}+3q^{-30}-11q^{-32}-6q^{-34}+2q^{-36}+11q^{-38}-8q^{-40}-3q^{-42}+23q^{-44}-13q^{-46}-9q^{-48}+22q^{-50}-11q^{-52}-10q^{-54}+13q^{-56}-3q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
1,0,0	$q^{-1}-q^{-3}+2q^{-5}-q^{-7}+3q^{-9}-q^{-11}+4q^{-13}+2q^{-17}-q^{-21}-q^{-23}-4q^{-25}+q^{-27}-3q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-q^{-37}+q^{-39}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}-q^{-4}+4q^{-8}-q^{-10}-3q^{-12}+6q^{-14}+4q^{-16}-6q^{-18}+2q^{-20}+16q^{-22}+2q^{-24}-7q^{-26}+15q^{-28}+19q^{-30}-16q^{-32}-12q^{-34}+13q^{-36}-11q^{-38}-31q^{-40}-3q^{-42}+7q^{-44}-16q^{-46}-5q^{-48}+22q^{-50}+6q^{-52}-10q^{-54}+16q^{-56}+17q^{-58}-13q^{-60}-6q^{-62}+15q^{-64}-2q^{-66}-16q^{-68}+9q^{-72}-4q^{-74}-7q^{-76}+4q^{-78}+4q^{-80}-2q^{-82}-q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}-q^{-4}+2q^{-6}+3q^{-12}-q^{-14}+4q^{-16}+3q^{-20}+q^{-22}-q^{-26}-2q^{-28}-2q^{-30}-4q^{-32}+q^{-34}-3q^{-36}+2q^{-38}+2q^{-44}-q^{-46}+q^{-48}$

B2 Invariants.

Weight	Invariant
0,1	$1-2q^{-2}+6q^{-4}-10q^{-6}+16q^{-8}-21q^{-10}+27q^{-12}-27q^{-14}+27q^{-16}-22q^{-18}+15q^{-20}-2q^{-22}-11q^{-24}+26q^{-26}-37q^{-28}+47q^{-30}-53q^{-32}+52q^{-34}-48q^{-36}+37q^{-38}-26q^{-40}+11q^{-42}+q^{-44}-13q^{-46}+21q^{-48}-26q^{-50}+27q^{-52}-24q^{-54}+21q^{-56}-15q^{-58}+10q^{-60}-7q^{-62}+4q^{-64}-2q^{-66}+q^{-68}$
1,0	$q^{2}-2q^{-2}-2q^{-4}+4q^{-6}+7q^{-8}-2q^{-10}-12q^{-12}-4q^{-14}+16q^{-16}+16q^{-18}-13q^{-20}-23q^{-22}+4q^{-24}+31q^{-26}+13q^{-28}-24q^{-30}-22q^{-32}+15q^{-34}+27q^{-36}-q^{-38}-25q^{-40}-8q^{-42}+16q^{-44}+9q^{-46}-16q^{-48}-15q^{-50}+10q^{-52}+14q^{-54}-9q^{-56}-20q^{-58}+5q^{-60}+23q^{-62}+3q^{-64}-23q^{-66}-9q^{-68}+23q^{-70}+20q^{-72}-15q^{-74}-26q^{-76}+4q^{-78}+28q^{-80}+9q^{-82}-19q^{-84}-18q^{-86}+5q^{-88}+17q^{-90}+4q^{-92}-9q^{-94}-8q^{-96}+q^{-98}+6q^{-100}+2q^{-102}-2q^{-104}-2q^{-106}+q^{-110}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{-2}-2q^{-4}+4q^{-6}-5q^{-8}+10q^{-10}-12q^{-12}+16q^{-14}-18q^{-16}+24q^{-18}-21q^{-20}+23q^{-22}-17q^{-24}+22q^{-26}-8q^{-28}+7q^{-30}+5q^{-32}-9q^{-34}+15q^{-36}-31q^{-38}+26q^{-40}-41q^{-42}+34q^{-44}-47q^{-46}+37q^{-48}-36q^{-50}+36q^{-52}-24q^{-54}+20q^{-56}-8q^{-58}+8q^{-60}+7q^{-62}-11q^{-64}+13q^{-66}-19q^{-68}+23q^{-70}-21q^{-72}+16q^{-74}-19q^{-76}+17q^{-78}-11q^{-80}+8q^{-82}-8q^{-84}+6q^{-86}-3q^{-88}+2q^{-90}-2q^{-92}+q^{-94}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}-2q^{-4}+6q^{-6}-10q^{-8}+13q^{-10}-11q^{-12}+q^{-14}+20q^{-16}-43q^{-18}+67q^{-20}-74q^{-22}+48q^{-24}+8q^{-26}-84q^{-28}+152q^{-30}-169q^{-32}+131q^{-34}-35q^{-36}-85q^{-38}+181q^{-40}-207q^{-42}+155q^{-44}-36q^{-46}-85q^{-48}+163q^{-50}-153q^{-52}+71q^{-54}+49q^{-56}-142q^{-58}+171q^{-60}-123q^{-62}+10q^{-64}+117q^{-66}-214q^{-68}+239q^{-70}-182q^{-72}+57q^{-74}+88q^{-76}-213q^{-78}+259q^{-80}-226q^{-82}+114q^{-84}+29q^{-86}-152q^{-88}+201q^{-90}-163q^{-92}+57q^{-94}+65q^{-96}-143q^{-98}+139q^{-100}-64q^{-102}-48q^{-104}+143q^{-106}-173q^{-108}+137q^{-110}-44q^{-112}-60q^{-114}+134q^{-116}-157q^{-118}+131q^{-120}-68q^{-122}+2q^{-124}+50q^{-126}-77q^{-128}+77q^{-130}-60q^{-132}+38q^{-134}-12q^{-136}-9q^{-138}+21q^{-140}-28q^{-142}+24q^{-144}-17q^{-146}+10q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

.

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["10 111"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+9t^{2}-17t+21-17t^{-1}+9t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-3z^{4}+z^{2}+1$

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]=

{ 77, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-3q^{9}+6q^{8}-10q^{7}+12q^{6}-13q^{5}+12q^{4}-9q^{3}+7q^{2}-3q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+10z^{8}a^{-6}+6z^{8}a^{-8}+3z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+7z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-4}-26z^{6}a^{-6}-11z^{6}a^{-8}+5z^{6}a^{-10}-8z^{5}a^{-3}-20z^{5}a^{-5}-28z^{5}a^{-7}-13z^{5}a^{-9}+3z^{5}a^{-11}-3z^{4}a^{-2}+3z^{4}a^{-4}+22z^{4}a^{-6}+10z^{4}a^{-8}-5z^{4}a^{-10}+z^{4}a^{-12}+5z^{3}a^{-3}+19z^{3}a^{-5}+30z^{3}a^{-7}+13z^{3}a^{-9}-3z^{3}a^{-11}+3z^{2}a^{-2}-2z^{2}a^{-4}-10z^{2}a^{-6}-3z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-za^{-3}-7za^{-5}-10za^{-7}-4za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$

Vassiliev invariants

V2 and V3:

(1, 0)

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 $4$ $0$ $8$ ${\frac {14}{3}}$ ${\frac {82}{3}}$ $0$ $64$ $96$ $64$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ ${\frac {328}{3}}$ ${\frac {9151}{30}}$ ${\frac {2258}{15}}$ ${\frac {19262}{45}}$ ${\frac {161}{18}}$ ${\frac {991}{30}}$

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 10 111. 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 21                     1 1 19                   2   -2 17                 4 1   3 15               6 2     -4 13             6 4       2 11           7 6         -1 9         5 6           -1 7       4 7             3 5     3 5               -2 3   1 5                 4 1   2                   -2 -1 1                     1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 111]]
Out[2]=	10
In[3]:=	PD[Knot[10, 111]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 15, 7], X[8, 14, 9, 13], X[2, 10, 3, 9], X[18, 12, 19, 11], X[16, 5, 17, 6], X[4, 17, 5, 18], X[20, 16, 1, 15], X[12, 20, 13, 19]]
In[4]:=	GaussCode[Knot[10, 111]]
Out[4]=	GaussCode[1, -5, 2, -8, 7, -1, 3, -4, 5, -2, 6, -10, 4, -3, 9, -7, 8, -6, 10, -9]
In[5]:=	BR[Knot[10, 111]]
Out[5]=	BR[4, {1, 1, 2, 2, -3, 2, 2, -1, 2, -3, 2}]
In[6]:=	alex = Alexander[Knot[10, 111]][t]
Out[6]=	2 9 17 2 3 21 - -- + -- - -- - 17 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 111]][z]
Out[7]=	2 4 6 1 + z - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 111]}
In[9]:=	{KnotDet[Knot[10, 111]], KnotSignature[Knot[10, 111]]}
Out[9]=	{77, 4}
In[10]:=	J=Jones[Knot[10, 111]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 7 q - 9 q + 12 q - 13 q + 12 q - 10 q + 6 q - 3 q + 10 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 111]}
In[12]:=	A2Invariant[Knot[10, 111]][q]
Out[12]=	2 4 6 8 10 12 14 18 20 22 1 - q + q + 2 q - q + 4 q - q + q - 3 q + q - 3 q + 24 26 28 30 q + q - q + q
In[13]:=	Kauffman[Knot[10, 111]][a, z]
Out[13]=	2 2 2 -8 3 2 -2 4 z 10 z 7 z z z z 3 z a + -- + -- - a - --- - ---- - --- - -- - --- + --- - ---- - 6 4 9 7 5 3 12 10 8 a a a a a a a a a 2 2 2 3 3 3 3 3 4 10 z 2 z 3 z 3 z 13 z 30 z 19 z 5 z z ----- - ---- + ---- - ---- + ----- + ----- + ----- + ---- + --- - 6 4 2 11 9 7 5 3 12 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 z 10 z 22 z 3 z 3 z 3 z 13 z 28 z 20 z ---- + ----- + ----- + ---- - ---- + ---- - ----- - ----- - ----- - 10 8 6 4 2 11 9 7 5 a a a a a a a a a 5 6 6 6 6 6 7 7 7 7 8 z 5 z 11 z 26 z 9 z z 7 z 7 z 3 z 3 z ---- + ---- - ----- - ----- - ---- + -- + ---- + ---- + ---- + ---- + 3 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 9 9 6 z 10 z 4 z 2 z 2 z ---- + ----- + ---- + ---- + ---- 8 6 4 7 5 a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 111]], Vassiliev[3][Knot[10, 111]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 111]][q, t]
Out[15]=	3 3 5 1 2 q q 5 7 7 2 9 2 5 q + 3 q + ---- + --- + -- + 5 q t + 4 q t + 7 q t + 5 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 6 q t + 7 q t + 6 q t + 6 q t + 4 q t + 6 q t + 15 6 17 6 17 7 19 7 21 8 2 q t + 4 q t + q t + 2 q t + q t