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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-8 + 4q-7 - 10q-6 + 16q-5 - 22q-4 + 26q-3 - 24q-2 + 23q-1 - 15 + 10q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 + q-22 - q-20 - 4q-18 + 2q-16 - 6q-14 + q-12 + 3q-10 + 2q-8 + 12q-6 + 2q-4 + 10q-2 + 3 + 4q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2z-2 + 4 + 4z2 + 3z4 + z6 - 5a2z-2 - 10a2 - 13a2z2 - 11a2z4 - 5a2z6 - a2z8 + 4a4z-2 + 8a4 + 10a4z2 + 7a4z4 + 2a4z6 - a6z-2 - 2a6 - 2a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 4a-1z3 - 8a-1z5 + 4a-1z7 - 2z-2 + 7 - 12z2 + 18z4 - 19z6 + 8z8 + 5az-1 - 11az + 14az3 - 9az5 - 7az7 + 7az9 - 5a2z-2 + 15a2 - 35a2z2 + 58a2z4 - 57a2z6 + 19a2z8 + 2a2z10 + 9a3z-1 - 24a3z + 36a3z3 - 23a3z5 - 13a3z7 + 14a3z9 - 4a4z-2 + 12a4 - 28a4z2 + 51a4z4 - 56a4z6 + 22a4z8 + 2a4z10 + 5a5z-1 - 18a5z + 38a5z3 - 37a5z5 + 7a5z7 + 7a5z9 - a6z-2 + 3a6 - 5a6z2 + 9a6z4 - 15a6z6 + 11a6z8 + a7z-1 - 5a7z + 11a7z3 - 14a7z5 + 9a7z7 + a8z2 - 4a8z4 + 4a8z6 - a9z3 + 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                 9 4     j = -1               14 6       j = -3             12 11         j = -5           14 12           j = -7         9 13             j = -9       7 13               j = -11     3 9                 j = -13   1 7                   j = -15   3                     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, 426]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 426]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[20, 11, 21, 12], > X[22, 17, 11, 18], X[16, 21, 17, 22], X[10, 13, 5, 14], X[8, 20, 9, 19], > X[18, 8, 19, 7], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, > {4, -2, 7, -3, 11, -6, 5, -9, 8, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-8 4 10 16 22 26 24 23 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]=	-24 -22 -20 4 2 6 -12 3 2 12 2 10 3 - q + q - q - --- + --- - --- + q + --- + -- + -- + -- + -- + 18 16 14 10 8 6 4 2 q q q q q q q q 4 6 8 > 4 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 426]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 4 - 10 a + 8 a - 2 a + -- - ---- + ---- - -- + 4 z - 13 a z + 10 a z - 2 2 2 2 z z z z 6 2 4 2 4 4 4 6 4 6 2 6 4 6 2 8 > 2 a z + 3 z - 11 a z + 7 a z - a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 426]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 7 + 15 a + 12 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 2 3 5 7 2 z 2 2 4 2 > 11 a z - 24 a z - 18 a z - 5 a z - 12 z + -- - 35 a z - 28 a z - 2 a 3 6 2 8 2 4 z 3 3 3 5 3 7 3 9 3 > 5 a z + a z + ---- + 14 a z + 36 a z + 38 a z + 11 a z - a z + a 4 5 4 2 z 2 4 4 4 6 4 8 4 8 z 5 > 18 z - ---- + 58 a z + 51 a z + 9 a z - 4 a z - ---- - 9 a z - 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 23 a z - 37 a z - 14 a z + a z - 19 z + -- - 57 a z - 56 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 15 a z + 4 a z + ---- - 7 a z - 13 a z + 7 a z + 9 a z + 8 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 19 a z + 22 a z + 11 a z + 7 a z + 14 a z + 7 a z + 2 a z + 4 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 14 1 3 1 7 3 9 7 13 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 9 13 14 12 12 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 4 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 L11a426

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