10 12

10_11

10_13

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10 12 Quick Notes

10 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X11,1,12,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -8, 9, -3, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 20 18 6 8 12
Conway Notation	[4312]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 9.8175 A-Polynomial See Data:10 12/A-polynomial

[edit Notes for 10 12's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-6t^{2}+10t-11+10t^{-1}-6t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+6z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-7q^{2}+6q-3+2q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+4z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+5z^{2}a^{-2}+5z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}+2a^{-2}+2a^{-4}-2a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+2z^{7}a^{-1}-z^{7}a^{-3}-z^{7}a^{-5}+2z^{7}a^{-7}-5z^{6}a^{-2}-14z^{6}a^{-4}-5z^{6}a^{-6}+2z^{6}a^{-8}+2z^{6}+az^{5}-4z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-7}+z^{5}a^{-9}+4z^{4}a^{-2}+23z^{4}a^{-4}+8z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-3az^{3}+5z^{3}a^{-3}+4z^{3}a^{-5}-z^{3}a^{-7}-3z^{3}a^{-9}+2z^{2}a^{-2}-12z^{2}a^{-4}-8z^{2}a^{-6}+2z^{2}a^{-8}+4z^{2}+az+za^{-1}-za^{-3}-3za^{-5}+2za^{-9}-2a^{-2}+2a^{-4}+2a^{-6}-1$
The A2 invariant	$-q^{6}+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-10}+2q^{-12}-q^{-14}+q^{-16}-q^{-18}-q^{-20}-q^{-24}$
The G2 invariant	$q^{32}-q^{30}+2q^{28}-3q^{26}+2q^{24}-2q^{22}-2q^{20}+6q^{18}-9q^{16}+9q^{14}-10q^{12}+6q^{10}-10q^{6}+17q^{4}-23q^{2}+23-15q^{-2}+q^{-4}+14q^{-6}-26q^{-8}+38q^{-10}-30q^{-12}+14q^{-14}+6q^{-16}-23q^{-18}+30q^{-20}-21q^{-22}+8q^{-24}+12q^{-26}-20q^{-28}+21q^{-30}-6q^{-32}-14q^{-34}+32q^{-36}-39q^{-38}+30q^{-40}-9q^{-42}-16q^{-44}+40q^{-46}-49q^{-48}+48q^{-50}-29q^{-52}+3q^{-54}+23q^{-56}-40q^{-58}+42q^{-60}-27q^{-62}+9q^{-64}+13q^{-66}-23q^{-68}+22q^{-70}-8q^{-72}-10q^{-74}+24q^{-76}-29q^{-78}+14q^{-80}+4q^{-82}-23q^{-84}+34q^{-86}-34q^{-88}+22q^{-90}-7q^{-92}-13q^{-94}+22q^{-96}-28q^{-98}+23q^{-100}-14q^{-102}+4q^{-104}+4q^{-106}-11q^{-108}+13q^{-110}-11q^{-112}+8q^{-114}-3q^{-116}-q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_12.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+q^{3}-q+3q^{-1}-q^{-3}+q^{-5}+q^{-7}-q^{-9}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{16}-q^{14}-q^{12}+2q^{10}-3q^{8}-2q^{6}+6q^{4}-2q^{2}-5+10q^{-2}-9q^{-6}+8q^{-8}+3q^{-10}-6q^{-12}+2q^{-14}+4q^{-16}-6q^{-20}+4q^{-22}+5q^{-24}-10q^{-26}+q^{-28}+8q^{-30}-8q^{-32}-2q^{-34}+7q^{-36}-3q^{-38}-2q^{-40}+3q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$-q^{33}+q^{31}+q^{29}-q^{25}+q^{23}+q^{21}-3q^{19}-2q^{17}+4q^{15}+3q^{13}-8q^{11}-7q^{9}+7q^{7}+15q^{5}-6q^{3}-19q-q^{-1}+29q^{-3}+9q^{-5}-28q^{-7}-18q^{-9}+27q^{-11}+27q^{-13}-21q^{-15}-26q^{-17}+14q^{-19}+25q^{-21}-5q^{-23}-21q^{-25}-2q^{-27}+16q^{-29}+9q^{-31}-11q^{-33}-19q^{-35}+6q^{-37}+25q^{-39}-32q^{-43}-6q^{-45}+31q^{-47}+14q^{-49}-29q^{-51}-20q^{-53}+21q^{-55}+24q^{-57}-13q^{-59}-21q^{-61}+3q^{-63}+18q^{-65}+2q^{-67}-12q^{-69}-3q^{-71}+6q^{-73}+3q^{-75}-3q^{-77}-q^{-79}+2q^{-81}-2q^{-85}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{56}-q^{54}-q^{52}-q^{48}+3q^{46}-q^{44}+q^{42}+2q^{40}-5q^{38}+2q^{36}-3q^{34}+4q^{32}+11q^{30}-7q^{28}-5q^{26}-16q^{24}+2q^{22}+27q^{20}+6q^{18}-2q^{16}-40q^{14}-23q^{12}+29q^{10}+37q^{8}+40q^{6}-41q^{4}-70q^{2}-21+40q^{-2}+113q^{-4}+16q^{-6}-89q^{-8}-99q^{-10}-19q^{-12}+143q^{-14}+97q^{-16}-40q^{-18}-132q^{-20}-92q^{-22}+103q^{-24}+124q^{-26}+21q^{-28}-96q^{-30}-107q^{-32}+35q^{-34}+88q^{-36}+50q^{-38}-39q^{-40}-79q^{-42}-15q^{-44}+44q^{-46}+58q^{-48}+5q^{-50}-51q^{-52}-63q^{-54}+10q^{-56}+78q^{-58}+51q^{-60}-32q^{-62}-116q^{-64}-33q^{-66}+90q^{-68}+106q^{-70}+12q^{-72}-143q^{-74}-87q^{-76}+55q^{-78}+127q^{-80}+78q^{-82}-102q^{-84}-112q^{-86}-23q^{-88}+81q^{-90}+113q^{-92}-21q^{-94}-71q^{-96}-65q^{-98}+3q^{-100}+78q^{-102}+27q^{-104}-4q^{-106}-44q^{-108}-33q^{-110}+22q^{-112}+15q^{-114}+21q^{-116}-6q^{-118}-18q^{-120}-6q^{-124}+10q^{-126}+4q^{-128}-2q^{-130}+3q^{-132}-7q^{-134}+4q^{-142}-q^{-144}-q^{-148}-q^{-150}+q^{-152}$
5	$-q^{85}+q^{83}+q^{81}+q^{77}-q^{75}-3q^{73}-q^{71}+q^{69}+4q^{65}+4q^{63}-3q^{61}-7q^{59}-5q^{57}-q^{55}+8q^{53}+16q^{51}+8q^{49}-13q^{47}-22q^{45}-17q^{43}+7q^{41}+34q^{39}+39q^{37}+4q^{35}-45q^{33}-61q^{31}-32q^{29}+33q^{27}+87q^{25}+76q^{23}-6q^{21}-94q^{19}-130q^{17}-67q^{15}+68q^{13}+171q^{11}+160q^{9}+26q^{7}-174q^{5}-268q^{3}-154q+106q^{-1}+332q^{-3}+336q^{-5}+34q^{-7}-350q^{-9}-478q^{-11}-219q^{-13}+263q^{-15}+583q^{-17}+426q^{-19}-124q^{-21}-595q^{-23}-574q^{-25}-54q^{-27}+524q^{-29}+655q^{-31}+223q^{-33}-403q^{-35}-650q^{-37}-324q^{-39}+247q^{-41}+566q^{-43}+376q^{-45}-109q^{-47}-447q^{-49}-364q^{-51}+8q^{-53}+312q^{-55}+316q^{-57}+64q^{-59}-200q^{-61}-265q^{-63}-107q^{-65}+115q^{-67}+220q^{-69}+142q^{-71}-54q^{-73}-208q^{-75}-196q^{-77}+4q^{-79}+224q^{-81}+261q^{-83}+60q^{-85}-241q^{-87}-366q^{-89}-142q^{-91}+255q^{-93}+465q^{-95}+258q^{-97}-220q^{-99}-553q^{-101}-399q^{-103}+145q^{-105}+589q^{-107}+528q^{-109}-5q^{-111}-554q^{-113}-618q^{-115}-158q^{-117}+429q^{-119}+647q^{-121}+316q^{-123}-255q^{-125}-579q^{-127}-419q^{-129}+45q^{-131}+440q^{-133}+455q^{-135}+116q^{-137}-260q^{-139}-392q^{-141}-224q^{-143}+79q^{-145}+283q^{-147}+250q^{-149}+45q^{-151}-148q^{-153}-207q^{-155}-111q^{-157}+37q^{-159}+135q^{-161}+121q^{-163}+25q^{-165}-60q^{-167}-89q^{-169}-54q^{-171}+8q^{-173}+51q^{-175}+50q^{-177}+16q^{-179}-18q^{-181}-32q^{-183}-22q^{-185}-3q^{-187}+16q^{-189}+20q^{-191}+7q^{-193}-4q^{-195}-9q^{-197}-10q^{-199}-4q^{-201}+6q^{-203}+6q^{-205}+2q^{-207}+2q^{-209}-2q^{-211}-4q^{-213}-q^{-215}+q^{-217}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-10}+2q^{-12}-q^{-14}+q^{-16}-q^{-18}-q^{-20}-q^{-24}$
1,1	$q^{20}-2q^{18}+4q^{16}-8q^{14}+15q^{12}-22q^{10}+28q^{8}-42q^{6}+55q^{4}-70q^{2}+78-96q^{-2}+105q^{-4}-98q^{-6}+90q^{-8}-56q^{-10}+29q^{-12}+30q^{-14}-76q^{-16}+128q^{-18}-171q^{-20}+202q^{-22}-216q^{-24}+214q^{-26}-199q^{-28}+162q^{-30}-118q^{-32}+64q^{-34}-19q^{-36}-32q^{-38}+70q^{-40}-92q^{-42}+100q^{-44}-102q^{-46}+96q^{-48}-80q^{-50}+66q^{-52}-54q^{-54}+40q^{-56}-28q^{-58}+19q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{18}-q^{14}+q^{10}-2q^{8}-4q^{6}+q^{4}+3q^{2}-2-2q^{-2}+5q^{-4}+2q^{-6}-3q^{-8}+q^{-10}+4q^{-12}+5q^{-18}+4q^{-20}-2q^{-22}+4q^{-24}+4q^{-26}-3q^{-28}-3q^{-30}+2q^{-32}-q^{-34}-6q^{-36}-3q^{-38}+q^{-40}-q^{-42}-3q^{-44}+q^{-46}+2q^{-48}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-q^{12}+q^{8}-3q^{6}+2q^{2}-6+7q^{-4}-7q^{-6}+2q^{-8}+10q^{-10}-2q^{-12}+q^{-14}+7q^{-16}+q^{-18}-q^{-20}+q^{-22}+4q^{-24}-2q^{-26}-6q^{-28}+5q^{-30}-q^{-32}-10q^{-34}+5q^{-36}+q^{-38}-7q^{-40}+3q^{-42}+q^{-44}-3q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$-q^{7}-q^{3}+q-q^{-1}+2q^{-3}-q^{-5}+2q^{-7}+q^{-9}+2q^{-11}+2q^{-13}+q^{-15}+2q^{-17}-q^{-19}+q^{-21}-2q^{-23}-2q^{-27}-q^{-31}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+q^{12}-2q^{10}+3q^{8}-5q^{6}+6q^{4}-8q^{2}+8-8q^{-2}+9q^{-4}-5q^{-6}+4q^{-8}+2q^{-10}-4q^{-12}+11q^{-14}-13q^{-16}+17q^{-18}-17q^{-20}+17q^{-22}-16q^{-24}+12q^{-26}-8q^{-28}+3q^{-30}+q^{-32}-4q^{-34}+7q^{-36}-9q^{-38}+9q^{-40}-9q^{-42}+7q^{-44}-5q^{-46}+4q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{24}-q^{20}-q^{18}+q^{16}+2q^{14}-q^{12}-4q^{10}-2q^{8}+3q^{6}+5q^{4}-2q^{2}-8-3q^{-2}+7q^{-4}+9q^{-6}-4q^{-8}-9q^{-10}-q^{-12}+10q^{-14}+6q^{-16}-4q^{-18}-5q^{-20}+5q^{-22}+7q^{-24}-5q^{-28}+q^{-30}+6q^{-32}+2q^{-34}-5q^{-36}-3q^{-38}+5q^{-40}+4q^{-42}-5q^{-44}-7q^{-46}+3q^{-48}+8q^{-50}-q^{-52}-10q^{-54}-6q^{-56}+7q^{-58}+8q^{-60}-3q^{-62}-9q^{-64}-3q^{-66}+6q^{-68}+4q^{-70}-2q^{-72}-4q^{-74}-q^{-76}+3q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{18}-q^{16}+q^{14}-2q^{12}+3q^{10}-4q^{8}+3q^{6}-6q^{4}+5q^{2}-8+5q^{-2}-7q^{-4}+7q^{-6}-5q^{-8}+4q^{-10}+q^{-12}+4q^{-14}+6q^{-16}-4q^{-18}+10q^{-20}-7q^{-22}+15q^{-24}-11q^{-26}+14q^{-28}-12q^{-30}+15q^{-32}-10q^{-34}+9q^{-36}-10q^{-38}+3q^{-40}-3q^{-42}-2q^{-44}-q^{-46}-6q^{-48}+5q^{-50}-7q^{-52}+6q^{-54}-8q^{-56}+7q^{-58}-6q^{-60}+4q^{-62}-4q^{-64}+4q^{-66}-2q^{-68}+2q^{-70}-q^{-72}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-q^{30}+2q^{28}-3q^{26}+2q^{24}-2q^{22}-2q^{20}+6q^{18}-9q^{16}+9q^{14}-10q^{12}+6q^{10}-10q^{6}+17q^{4}-23q^{2}+23-15q^{-2}+q^{-4}+14q^{-6}-26q^{-8}+38q^{-10}-30q^{-12}+14q^{-14}+6q^{-16}-23q^{-18}+30q^{-20}-21q^{-22}+8q^{-24}+12q^{-26}-20q^{-28}+21q^{-30}-6q^{-32}-14q^{-34}+32q^{-36}-39q^{-38}+30q^{-40}-9q^{-42}-16q^{-44}+40q^{-46}-49q^{-48}+48q^{-50}-29q^{-52}+3q^{-54}+23q^{-56}-40q^{-58}+42q^{-60}-27q^{-62}+9q^{-64}+13q^{-66}-23q^{-68}+22q^{-70}-8q^{-72}-10q^{-74}+24q^{-76}-29q^{-78}+14q^{-80}+4q^{-82}-23q^{-84}+34q^{-86}-34q^{-88}+22q^{-90}-7q^{-92}-13q^{-94}+22q^{-96}-28q^{-98}+23q^{-100}-14q^{-102}+4q^{-104}+4q^{-106}-11q^{-108}+13q^{-110}-11q^{-112}+8q^{-114}-3q^{-116}-q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-6t^{2}+10t-11+10t^{-1}-6t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}a^{-4}+4z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+5z^{2}a^{-2}+5z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}+2a^{-2}+2a^{-4}-2a^{-6}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+2z^{7}a^{-1}-z^{7}a^{-3}-z^{7}a^{-5}+2z^{7}a^{-7}-5z^{6}a^{-2}-14z^{6}a^{-4}-5z^{6}a^{-6}+2z^{6}a^{-8}+2z^{6}+az^{5}-4z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-7}+z^{5}a^{-9}+4z^{4}a^{-2}+23z^{4}a^{-4}+8z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-3az^{3}+5z^{3}a^{-3}+4z^{3}a^{-5}-z^{3}a^{-7}-3z^{3}a^{-9}+2z^{2}a^{-2}-12z^{2}a^{-4}-8z^{2}a^{-6}+2z^{2}a^{-8}+4z^{2}+az+za^{-1}-za^{-3}-3za^{-5}+2za^{-9}-2a^{-2}+2a^{-4}+2a^{-6}-1$

Vassiliev invariants

V2 and V3:

(4, 6)

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 $16$ $48$ $128$ ${\frac {680}{3}}$ ${\frac {40}{3}}$ $768$ $1152$ $192$ $80$ ${\frac {2048}{3}}$ $1152$ ${\frac {10880}{3}}$ ${\frac {640}{3}}$ ${\frac {92822}{15}}$ ${\frac {12232}{15}}$ ${\frac {59048}{45}}$ ${\frac {538}{9}}$ ${\frac {422}{15}}$

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=$2 is the signature of 10 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-3	-2	-1	0	1	2	3	4	5	6	7	χ
17											1	-1
15										1		1
13									3	1		-2
11								3	1			2
9							4	3				-1
7						4	3					1
5					3	4						1
3				3	4							-1
1			1	4								3
-1		1	2									-1
-3		1										1
-5	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, 12]]
Out[2]=	10
In[3]:=	PD[Knot[10, 12]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[13, 19, 14, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[11, 1, 12, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 12]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -8, 9, -3, 4, -6, 5, -7, 3, -9, 8]
In[5]:=	BR[Knot[10, 12]]
Out[5]=	BR[4, {1, 1, 1, 1, 1, 2, -1, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 12]][t]
Out[6]=	2 6 10 2 3 -11 + -- - -- + -- + 10 t - 6 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 12]][z]
Out[7]=	2 4 6 1 + 4 z + 6 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 12], Knot[10, 54]}
In[9]:=	{KnotDet[Knot[10, 12]], KnotSignature[Knot[10, 12]]}
Out[9]=	{47, 2}
In[10]:=	J=Jones[Knot[10, 12]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 -3 - q + - + 6 q - 7 q + 8 q - 7 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 12]}
In[12]:=	A2Invariant[Knot[10, 12]][q]
Out[12]=	-6 2 4 6 8 10 12 14 16 18 20 24 -q + 2 q - q + 2 q + q + q + 2 q - q + q - q - q - q
In[13]:=	Kauffman[Knot[10, 12]][a, z]
Out[13]=	2 2 2 2 2 2 z 3 z z z 2 2 z 8 z -1 + -- + -- - -- + --- - --- - -- + - + a z + 4 z + ---- - ---- - 6 4 2 9 5 3 a 8 6 a a a a a a a a 2 2 3 3 3 3 4 12 z 2 z 3 z z 4 z 5 z 3 4 5 z ----- + ---- - ---- - -- + ---- + ---- - 3 a z - 6 z - ---- + 4 2 9 7 5 3 8 a a a a a a a 4 4 4 5 5 5 5 6 8 z 23 z 4 z z 3 z z 4 z 5 6 2 z ---- + ----- + ---- + -- - ---- - -- - ---- + a z + 2 z + ---- - 6 4 2 9 7 3 a 8 a a a a a a a 6 6 6 7 7 7 7 8 8 8 5 z 14 z 5 z 2 z z z 2 z 2 z 4 z 2 z ---- - ----- - ---- + ---- - -- - -- + ---- + ---- + ---- + ---- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 9 9 z z -- + -- 5 3 a a
In[14]:=	{Vassiliev[2][Knot[10, 12]], Vassiliev[3][Knot[10, 12]]}
Out[14]=	{0, 6}
In[15]:=	Kh[Knot[10, 12]][q, t]
Out[15]=	3 1 1 1 2 q 3 5 4 q + 3 q + ----- + ----- + ---- + --- + - + 4 q t + 3 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 11 5 4 q t + 4 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 13 5 13 6 15 6 17 7 3 q t + q t + q t + q t