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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X10,20,5,19

Gauss Code:

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

Jones Polynomial:

q-6 - 5q-5 + 10q-4 - 17q-3 + 23q-2 - 25q-1 + 27 - 21q + 17q2 - 9q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 - 2q-16 - q-14 - 7q-10 + 2q-8 - 2q-6 + 2q-4 + 9q-2 + 3 + 13q2 + 2q4 + 5q6 + 4q8 - 3q10 + 2q12 - q14

HOMFLY-PT Polynomial:

2a-2z-2 + a-2 - 3a-2z2 - 3a-2z4 - a-2z6 - 5z-2 - 4 + 6z2 + 10z4 + 5z6 + z8 + 4a2z-2 + 4a2 - 4a2z2 - 6a2z4 - 2a2z6 - a4z-2 - a4 + a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 5a-4z4 + 4a-4z6 + 4a-3z3 - 10a-3z5 + 8a-3z7 + 2a-2z-2 - 3a-2 - 5a-2z2 + 14a-2z4 - 17a-2z6 + 11a-2z8 - 5a-1z-1 + 5a-1z + 10a-1z3 - 15a-1z5 - a-1z7 + 8a-1z9 + 5z-2 - 4 - 21z2 + 59z4 - 64z6 + 24z8 + 2z10 - 9az-1 + 9az + 17az3 - 19az5 - 15az7 + 15az9 + 4a2z-2 - 2a2 - 18a2z2 + 53a2z4 - 63a2z6 + 22a2z8 + 2a2z10 - 5a3z-1 + 5a3z + 17a3z3 - 25a3z5 - a3z7 + 7a3z9 + a4z-2 - 4a4z2 + 12a4z4 - 19a4z6 + 9a4z8 - a5z-1 + a5z + 5a5z3 - 10a5z5 + 5a5z7 - a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   6 1   j = 5                 11 3     j = 3               12 8       j = 1             15 9         j = -1           12 14           j = -3         11 13             j = -5       6 12               j = -7     4 11                 j = -9   1 6                   j = -11   4                     j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 398]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 398]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[18, 10, 19, 9], X[22, 17, 11, 18], X[20, 11, 21, 12], > X[16, 21, 17, 22], X[4, 15, 1, 16], X[10, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, > {8, -2, 4, -5, 10, -9, 7, -6, 11, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-6 5 10 17 23 25 2 3 4 5 27 + q - -- + -- - -- + -- - -- - 21 q + 17 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 7 2 2 2 9 2 4 6 8 3 + q - --- - q - --- + -- - -- + -- + -- + 13 q + 2 q + 5 q + 4 q - 16 10 8 6 4 2 q q q q q q 10 12 14 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 398]][a, z]
Out[8]=	2 4 2 -2 2 4 5 2 4 a a 2 3 z 2 2 4 2 -4 + a + 4 a - a - -- + ----- + ---- - -- + 6 z - ---- - 4 a z + a z + 2 2 2 2 2 2 z a z z z a 4 6 4 3 z 2 4 4 4 6 z 2 6 8 > 10 z - ---- - 6 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 398]][a, z]
Out[9]=	2 4 3 5 3 2 5 2 4 a a 5 9 a 5 a a 5 z -4 - -- - 2 a + -- + ----- + ---- + -- - --- - --- - ---- - -- + --- + 9 a z + 2 2 2 2 2 2 a z z z z a a z a z z z 2 2 3 3 3 5 2 2 z 5 z 2 2 4 2 z 4 z > 5 a z + a z - 21 z + ---- - ---- - 18 a z - 4 a z - -- + ---- + 4 2 5 3 a a a a 3 4 4 10 z 3 3 3 5 3 4 5 z 14 z 2 4 > ----- + 17 a z + 17 a z + 5 a z + 59 z - ---- + ----- + 53 a z + a 4 2 a a 5 5 5 4 4 6 4 z 10 z 15 z 5 3 5 5 5 > 12 a z - a z + -- - ----- - ----- - 19 a z - 25 a z - 10 a z - 5 3 a a a 6 6 7 7 6 4 z 17 z 2 6 4 6 6 6 8 z z 7 > 64 z + ---- - ----- - 63 a z - 19 a z + a z + ---- - -- - 15 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 11 z 2 8 4 8 8 z 9 > a z + 5 a z + 24 z + ----- + 22 a z + 9 a z + ---- + 15 a z + 2 a a 3 9 10 2 10 > 7 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 4 1 6 4 11 6 12 -- + 15 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 11 13 12 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 12 q t + 8 q t + 11 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a398

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