10 59

10_58

10_60

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10 59 Quick Notes

Two figure 8 knots on a loop, interlinked.

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code	1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 12 16
Conway Notation	[22,211,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 13.3899 A-Polynomial See Data:10 59/A-polynomial

[edit Notes for 10 59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for 10 59's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}+a^{2}+4a^{-2}-3a^{-4}+a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$
The A2 invariant	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_59.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+3q^{3}-3q+3q^{-1}+2q^{-7}-4q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-2q^{18}+8q^{16}-3q^{14}-12q^{12}+17q^{10}+4q^{8}-26q^{6}+15q^{4}+17q^{2}-27+4q^{-2}+22q^{-4}-14q^{-6}-9q^{-8}+14q^{-10}+6q^{-12}-16q^{-14}-3q^{-16}+24q^{-18}-15q^{-20}-18q^{-22}+29q^{-24}-5q^{-26}-19q^{-28}+17q^{-30}+q^{-32}-9q^{-34}+5q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-2q^{41}+3q^{39}+8q^{37}-3q^{35}-19q^{33}-q^{31}+32q^{29}+16q^{27}-45q^{25}-44q^{23}+50q^{21}+79q^{19}-35q^{17}-115q^{15}+3q^{13}+142q^{11}+42q^{9}-147q^{7}-91q^{5}+129q^{3}+134q-102q^{-1}-154q^{-3}+61q^{-5}+165q^{-7}-19q^{-9}-156q^{-11}-18q^{-13}+137q^{-15}+54q^{-17}-106q^{-19}-87q^{-21}+62q^{-23}+119q^{-25}-17q^{-27}-137q^{-29}-42q^{-31}+145q^{-33}+91q^{-35}-128q^{-37}-134q^{-39}+103q^{-41}+152q^{-43}-64q^{-45}-143q^{-47}+23q^{-49}+121q^{-51}-q^{-53}-85q^{-55}-11q^{-57}+51q^{-59}+12q^{-61}-27q^{-63}-8q^{-65}+14q^{-67}+2q^{-69}-5q^{-71}+2q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-2q^{74}-2q^{72}+3q^{70}+3q^{68}+8q^{66}-10q^{64}-19q^{62}-q^{60}+14q^{58}+53q^{56}-q^{54}-67q^{52}-64q^{50}-13q^{48}+157q^{46}+117q^{44}-58q^{42}-213q^{40}-229q^{38}+174q^{36}+366q^{34}+219q^{32}-227q^{30}-624q^{28}-168q^{26}+432q^{24}+724q^{22}+204q^{20}-792q^{18}-775q^{16}-15q^{14}+977q^{12}+909q^{10}-399q^{8}-1123q^{6}-743q^{4}+676q^{2}+1331+282q^{-2}-940q^{-4}-1202q^{-6}+105q^{-8}+1243q^{-10}+766q^{-12}-499q^{-14}-1218q^{-16}-332q^{-18}+871q^{-20}+921q^{-22}-84q^{-24}-976q^{-26}-609q^{-28}+410q^{-30}+928q^{-32}+334q^{-34}-601q^{-36}-855q^{-38}-183q^{-40}+800q^{-42}+823q^{-44}+3q^{-46}-971q^{-48}-901q^{-50}+374q^{-52}+1141q^{-54}+764q^{-56}-682q^{-58}-1373q^{-60}-296q^{-62}+941q^{-64}+1238q^{-66}-68q^{-68}-1213q^{-70}-735q^{-72}+345q^{-74}+1079q^{-76}+379q^{-78}-622q^{-80}-646q^{-82}-105q^{-84}+558q^{-86}+371q^{-88}-157q^{-90}-293q^{-92}-175q^{-94}+169q^{-96}+168q^{-98}-8q^{-100}-65q^{-102}-83q^{-104}+34q^{-106}+40q^{-108}-2q^{-110}-21q^{-114}+7q^{-116}+5q^{-118}-4q^{-120}+4q^{-122}-3q^{-124}+2q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-2q^{113}-2q^{111}+3q^{109}+3q^{107}+3q^{105}+q^{103}-10q^{101}-19q^{99}-q^{97}+23q^{95}+35q^{93}+27q^{91}-23q^{89}-85q^{87}-89q^{85}+9q^{83}+138q^{81}+196q^{79}+96q^{77}-158q^{75}-378q^{73}-322q^{71}+72q^{69}+543q^{67}+695q^{65}+268q^{63}-568q^{61}-1177q^{59}-914q^{57}+266q^{55}+1540q^{53}+1828q^{51}+587q^{49}-1513q^{47}-2818q^{45}-1968q^{43}+793q^{41}+3448q^{39}+3691q^{37}+794q^{35}-3334q^{33}-5328q^{31}-3066q^{29}+2184q^{27}+6311q^{25}+5617q^{23}+15q^{21}-6259q^{19}-7859q^{17}-2875q^{15}+5033q^{13}+9235q^{11}+5829q^{9}-2791q^{7}-9491q^{5}-8329q^{3}+86q+8648q^{-1}+9869q^{-3}+2601q^{-5}-6974q^{-7}-10438q^{-9}-4742q^{-11}+4983q^{-13}+10058q^{-15}+6127q^{-17}-3010q^{-19}-9090q^{-21}-6793q^{-23}+1407q^{-25}+7849q^{-27}+6891q^{-29}-169q^{-31}-6607q^{-33}-6724q^{-35}-809q^{-37}+5491q^{-39}+6538q^{-41}+1725q^{-43}-4415q^{-45}-6478q^{-47}-2842q^{-49}+3226q^{-51}+6503q^{-53}+4267q^{-55}-1664q^{-57}-6445q^{-59}-5982q^{-61}-366q^{-63}+5954q^{-65}+7714q^{-67}+2969q^{-69}-4836q^{-71}-9102q^{-73}-5758q^{-75}+2879q^{-77}+9654q^{-79}+8445q^{-81}-279q^{-83}-9174q^{-85}-10342q^{-87}-2584q^{-89}+7509q^{-91}+11130q^{-93}+5151q^{-95}-5068q^{-97}-10600q^{-99}-6856q^{-101}+2367q^{-103}+8890q^{-105}+7465q^{-107}+29q^{-109}-6579q^{-111}-6955q^{-113}-1623q^{-115}+4154q^{-117}+5651q^{-119}+2375q^{-121}-2159q^{-123}-4050q^{-125}-2356q^{-127}+802q^{-129}+2556q^{-131}+1882q^{-133}-61q^{-135}-1413q^{-137}-1291q^{-139}-223q^{-141}+697q^{-143}+767q^{-145}+230q^{-147}-284q^{-149}-397q^{-151}-173q^{-153}+101q^{-155}+194q^{-157}+89q^{-159}-34q^{-161}-73q^{-163}-41q^{-165}+4q^{-167}+27q^{-169}+22q^{-171}-5q^{-173}-12q^{-175}-2q^{-179}-q^{-181}+5q^{-183}+q^{-185}-3q^{-187}+2q^{-189}-2q^{-193}+q^{-195}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
2,0	$q^{28}-2q^{24}-2q^{22}+3q^{20}+5q^{18}-4q^{16}-6q^{14}+6q^{12}+9q^{10}-6q^{8}-11q^{6}+7q^{4}+12q^{2}-9-9q^{-2}+11q^{-4}+3q^{-6}-7q^{-8}+7q^{-12}-2q^{-14}+10q^{-18}-3q^{-20}-12q^{-22}+5q^{-24}+8q^{-26}-15q^{-28}-8q^{-30}+13q^{-32}+8q^{-34}-8q^{-36}-4q^{-38}+10q^{-40}+3q^{-42}-6q^{-44}-2q^{-46}+q^{-48}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+2q^{16}+3q^{14}-8q^{12}+7q^{10}+5q^{8}-16q^{6}+13q^{4}+6q^{2}-21+14q^{-2}+9q^{-4}-18q^{-6}+8q^{-8}+11q^{-10}-5q^{-12}-4q^{-14}+3q^{-16}+7q^{-18}-13q^{-20}-7q^{-22}+20q^{-24}-13q^{-26}-10q^{-28}+24q^{-30}-9q^{-32}-10q^{-34}+15q^{-36}-3q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{13}+q^{9}-q^{7}+2q^{5}-3q^{3}+q-2q^{-1}+2q^{-3}+3q^{-7}+3q^{-9}+q^{-13}-3q^{-15}-4q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-2q^{18}+6q^{16}-9q^{14}+14q^{12}-19q^{10}+23q^{8}-26q^{6}+25q^{4}-22q^{2}+13-4q^{-2}-7q^{-4}+22q^{-6}-32q^{-8}+45q^{-10}-47q^{-12}+50q^{-14}-45q^{-16}+37q^{-18}-27q^{-20}+13q^{-22}-2q^{-24}-11q^{-26}+18q^{-28}-24q^{-30}+25q^{-32}-24q^{-34}+21q^{-36}-15q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{34}-2q^{30}-2q^{28}+4q^{26}+6q^{24}-3q^{22}-11q^{20}-3q^{18}+15q^{16}+13q^{14}-12q^{12}-22q^{10}+2q^{8}+27q^{6}+12q^{4}-23q^{2}-22+11q^{-2}+26q^{-4}+2q^{-6}-23q^{-8}-8q^{-10}+17q^{-12}+13q^{-14}-12q^{-16}-11q^{-18}+10q^{-20}+15q^{-22}-7q^{-24}-18q^{-26}+2q^{-28}+19q^{-30}+2q^{-32}-22q^{-34}-11q^{-36}+19q^{-38}+19q^{-40}-13q^{-42}-25q^{-44}+2q^{-46}+27q^{-48}+11q^{-50}-17q^{-52}-19q^{-54}+5q^{-56}+18q^{-58}+5q^{-60}-9q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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 59"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$

Vassiliev invariants

V2 and V3:

(-1, -1)

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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ ${\frac {38}{3}}$ $32$ ${\frac {112}{3}}$ ${\frac {64}{3}}$ $24$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {4769}{30}}$ ${\frac {194}{5}}$ ${\frac {1738}{45}}$ ${\frac {319}{18}}$ $-{\frac {31}{30}}$

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=$2 is the signature of 10 59. 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
15											1	1
13										2		-2
11									4	1		3
9								6	2			-4
7							6	4				2
5						6	6					0
3					6	6						0
1				4	7							3
-1			2	5								-3
-3		1	4									3
-5		2										-2
-7	1											1

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, 59]]
Out[2]=	10
In[3]:=	PD[Knot[10, 59]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16], X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[10, 59]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7]
In[5]:=	BR[Knot[10, 59]]
Out[5]=	BR[5, {-1, 2, -1, 2, -3, 2, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 59]][t]
Out[6]=	-3 7 18 2 3 -23 + t - -- + -- + 18 t - 7 t + t 2 t t
In[7]:=	Conway[Knot[10, 59]][z]
Out[7]=	2 4 6 1 - z - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}
In[9]:=	{KnotDet[Knot[10, 59]], KnotSignature[Knot[10, 59]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[10, 59]][q]
Out[10]=	-3 3 6 2 3 4 5 6 7 -9 + q - -- + - + 12 q - 12 q + 12 q - 10 q + 6 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 59], Knot[10, 106]}
In[12]:=	A2Invariant[Knot[10, 59]][q]
Out[12]=	-10 -6 2 2 2 4 6 8 10 12 14 q - q + -- - -- + 2 q - q + 4 q - q + q - q - 3 q + 4 2 q q 16 18 22 2 q - q + q
In[13]:=	Kauffman[Knot[10, 59]][a, z]
Out[13]=	2 2 -6 3 4 2 z 4 z 5 z 2 z 3 z -2 - a - -- - -- - a + -- - --- - --- - 2 a z + 8 z - -- + ---- + 4 2 7 3 a 8 6 a a a a a a 2 2 3 3 3 3 10 z 11 z 2 2 3 z 4 z 20 z 21 z 3 ----- + ----- + 3 a z - ---- + ---- + ----- + ----- + 8 a z - 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 4 z 5 z 11 z 8 z 2 4 3 z 7 z 28 z 6 z + -- - ---- - ----- - ---- - 3 a z + ---- - ---- - ----- - 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 7 7 27 z 5 6 5 z z 9 z 2 6 6 z 11 z ----- - 9 a z - 4 z + ---- + -- - ---- + a z + ---- + ----- + a 6 4 2 5 3 a a a a a 7 8 8 9 9 8 z 7 8 4 z 7 z z z ---- + 3 a z + 3 z + ---- + ---- + -- + -- a 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 59]], Vassiliev[3][Knot[10, 59]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[10, 59]][q, t]
Out[15]=	3 1 2 1 4 2 5 4 q 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 6 q t + 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 11 4 11 5 13 5 15 6 4 q t + q t + 2 q t + q t