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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,12,19,11 X16,8,17,7 X8,16,9,15 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 13q-3 + 19q-2 - 21q-1 + 23 - 19q + 15q2 - 9q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + 2q-14 + 3q-12 - 2q-10 + 7q-8 + 3q-6 + 3q-4 + 7q-2 + 6q2 - 3q4 + 3q8 - 4q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z2 + 2a-2z4 + z-2 + 2 - z2 - z4 - z6 - 2a2z-2 - 2a2 + 2a2z2 + 3a2z4 + a4z-2 - a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 5a-4z4 + 4a-4z6 - 2a-3z + 7a-3z3 - 12a-3z5 + 8a-3z7 + 2a-2 - 4a-2z2 + 5a-2z4 - 11a-2z6 + 9a-2z8 - 7a-1z + 27a-1z3 - 33a-1z5 + 9a-1z7 + 5a-1z9 - z-2 + 10 - 28z2 + 46z4 - 48z6 + 20z8 + z10 + 2az-1 - 14az + 34az3 - 31az5 - az7 + 9az9 - 2a2z-2 + 12a2 - 30a2z2 + 46a2z4 - 44a2z6 + 16a2z8 + a2z10 + 2a3z-1 - 10a3z + 20a3z3 - 18a3z5 + a3z7 + 4a3z9 - a4z-2 + 4a4 - 5a4z2 + 7a4z4 - 10a4z6 + 5a4z8 - a5z + 5a5z3 - 7a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + 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                 9 3     j = 3               11 7       j = 1             12 8         j = -1           10 12           j = -3         9 11             j = -5       5 11               j = -7     3 8                 j = -9   1 6                   j = -11   2                     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, 467]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 467]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[20, 13, 21, 14], X[22, 20, 15, 19], X[12, 21, 13, 22], > X[14, 18, 5, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, 9, -3, 7, -6, 8, -7}, > {10, -1, 4, -5, 11, -2, 3, -8, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 8 13 19 21 2 3 4 5 23 + q - -- + -- - -- + -- - -- - 19 q + 15 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 2 3 2 7 3 3 7 2 4 8 q + q - --- + --- + --- - --- + -- + -- + -- + -- + 6 q - 3 q + 3 q - 16 14 12 10 8 6 4 2 q q q q q q q q 10 12 14 16 > 4 q + 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 467]][a, z]
Out[8]=	2 4 2 2 2 4 6 -2 2 a a 2 z z 2 2 4 2 4 2 - 2 a - a + a + z - ---- + -- - z - -- + -- + 2 a z - 3 a z - z + 2 2 4 2 z z a a 4 2 z 2 4 6 > ---- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 467]][a, z]
Out[9]=	2 4 3 2 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z 10 + -- + 12 a + 4 a - a - z - ---- - -- + --- + ---- - --- - --- - 2 2 2 z z 3 a a z z a 2 2 3 5 2 2 z 4 z 2 2 4 2 > 14 a z - 10 a z - a z - 28 z + ---- - ---- - 30 a z - 5 a z + 4 2 a a 3 3 3 4 6 2 z 7 z 27 z 3 3 3 5 3 4 5 z > 3 a z - -- + ---- + ----- + 34 a z + 20 a z + 5 a z + 46 z - ---- + 5 3 a 4 a a a 4 5 5 5 5 z 2 4 4 4 6 4 z 12 z 33 z 5 > ---- + 46 a z + 7 a z - 3 a z + -- - ----- - ----- - 31 a z - 2 5 3 a a a a 6 6 3 5 5 5 6 4 z 11 z 2 6 4 6 6 6 > 18 a z - 7 a z - 48 z + ---- - ----- - 44 a z - 10 a z + a z + 4 2 a a 7 7 8 8 z 9 z 7 3 7 5 7 8 9 z 2 8 4 8 > ---- + ---- - a z + a z + 3 a z + 20 z + ---- + 16 a z + 5 a z + 3 a 2 a a 9 5 z 9 3 9 10 2 10 > ---- + 9 a z + 4 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 2 1 6 3 8 5 11 -- + 12 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 9 11 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 11 q t + 7 q t + 9 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 L11a467

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