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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 14q-5 - 19q-4 + 23q-3 - 22q-2 + 21q-1 - 15 + 10q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 - q-20 - q-18 + 6q-16 - 2q-14 + 3q-12 + 4q-10 + 7q-6 - q-4 + 7q-2 + 3 - q2 + 5q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + z-2 + 3 - 2z4 - 2a2z-2 - 5a2 - 3a2z2 + a2z6 + a4z-2 + a4 - 2a4z2 - 3a4z4 + 2a6 + 3a6z2 - a8

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 4a-1z3 - 8a-1z5 + 4a-1z7 - z-2 + 6 - 11z2 + 16z4 - 19z6 + 8z8 + 2az-1 - 5az + 6az3 - 5az5 - 9az7 + 7az9 - 2a2z-2 + 13a2 - 39a2z2 + 59a2z4 - 53a2z6 + 16a2z8 + 2a2z10 + 2a3z-1 - 8a3z + 10a3z3 + 3a3z5 - 21a3z7 + 13a3z9 - a4z-2 + 9a4 - 30a4z2 + 52a4z4 - 45a4z6 + 15a4z8 + 2a4z10 - 3a5z + 10a5z3 - 6a5z5 - 3a5z7 + 6a5z9 + 6a6z4 - 9a6z6 + 7a6z8 + a7z - 5a7z5 + 5a7z7 - a8 + 3a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   7 1   j = 1                 8 3     j = -1               13 7       j = -3             12 11         j = -5           11 10           j = -7         8 12             j = -9       6 11               j = -11     2 9                 j = -13   1 5                   j = -15   2                     j = -17 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, 457]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 457]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[18, 8, 19, 7], > X[22, 9, 13, 10], X[16, 21, 17, 22], X[8, 18, 9, 17], X[10, 15, 11, 16], > X[12, 20, 5, 19], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 5, -8, 3, -9}, > {11, -2, 8, -6, 7, -4, 9, -3, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 7 14 19 23 22 21 2 3 -15 - q + -- - -- + -- - -- + -- - -- + -- + 10 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 -20 -18 6 2 3 4 7 -4 7 3 - q - q + --- - q - q + --- - --- + --- + --- + -- - q + -- - 22 16 14 12 10 6 2 q q q q q q q 2 4 6 8 10 > q + 5 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 457]][a, z]
Out[8]=	2 4 2 2 4 6 8 -2 2 a a z 2 2 4 2 3 - 5 a + a + 2 a - a + z - ---- + -- + -- - 3 a z - 2 a z + 2 2 2 z z a 6 2 4 4 4 2 6 > 3 a z - 2 z - 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 457]][a, z]
Out[9]=	2 4 3 2 4 8 -2 2 a a 2 a 2 a 3 6 + 13 a + 9 a - a - z - ---- - -- + --- + ---- - 5 a z - 8 a z - 2 2 z z z z 2 3 5 7 9 2 z 2 2 4 2 8 2 4 z > 3 a z + a z + a z - 11 z + -- - 39 a z - 30 a z + 3 a z + ---- + 2 a a 4 3 3 3 5 3 9 3 4 2 z 2 4 > 6 a z + 10 a z + 10 a z - 2 a z + 16 z - ---- + 59 a z + 2 a 5 4 4 6 4 8 4 8 z 5 3 5 5 5 > 52 a z + 6 a z - 5 a z - ---- - 5 a z + 3 a z - 6 a z - a 6 7 5 9 5 6 z 2 6 4 6 6 6 8 6 > 5 a z + a z - 19 z + -- - 53 a z - 45 a z - 9 a z + 3 a z + 2 a 7 4 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- - 9 a z - 21 a z - 3 a z + 5 a z + 8 z + 16 a z + 15 a z + a 6 8 9 3 9 5 9 2 10 4 10 > 7 a z + 7 a z + 13 a z + 6 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 13 1 2 1 5 2 9 6 11 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 8 12 11 10 12 7 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 7 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a457

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