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DrawMorseLink

PD Presentation:

X12,1,13,2 X14,3,15,4 X18,10,19,9 X16,6,17,5 X22,7,11,8 X6,21,7,22 X20,15,21,16 X8,18,9,17 X4,20,5,19 X2,11,3,12 X10,13,1,14

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 13q-9/2 + 17q-7/2 - 21q-5/2 + 21q-3/2 - 19q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - 2q-18 + 4q-14 - 2q-12 + 5q-10 + 3q-4 - 3q-2 + 5 - 2q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - az-1 - 2az + 2az3 + 3az5 + az7 + a3z-1 + 5a3z + 7a3z3 + 4a3z5 + a3z7 - 3a5z - 3a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 13a-1z5 - 8a-1z7 + 2z2 - 6z4 + 13z6 - 9z8 + az-1 - az - 7az3 + 11az5 + az7 - 6az9 - a2 + 6a2z2 - 22a2z4 + 28a2z6 - 11a2z8 - 2a2z10 + a3z-1 - 5a3z + 9a3z3 - 16a3z5 + 21a3z7 - 11a3z9 + 8a4z2 - 23a4z4 + 24a4z6 - 7a4z8 - 2a4z10 + 2a5z3 - 5a5z5 + 9a5z7 - 5a5z9 + 3a6z2 - 9a6z4 + 12a6z6 - 5a6z8 + 2a7z - 6a7z3 + 8a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

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 = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 3     j = 0               11 6       j = -2             11 9         j = -4           10 10           j = -6         8 12             j = -8       5 9               j = -10     2 8                 j = -12   1 5                   j = -14   2                     j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 373]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 373]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[18, 10, 19, 9], X[16, 6, 17, 5], > X[22, 7, 11, 8], X[6, 21, 7, 22], X[20, 15, 21, 16], X[8, 18, 9, 17], > X[4, 20, 5, 19], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -6, 5, -8, 3, -11}, > {10, -1, 11, -2, 7, -4, 8, -3, 9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 13 17 21 21 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 4 2 5 3 3 2 6 8 10 5 - q + q - --- + --- - --- + --- + -- - -- - 2 q + 2 q - 2 q + q 18 14 12 10 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 373]][a, z]
Out[8]=	3 3 a a z 3 5 2 z 3 3 3 5 3 -(-) + -- - - - 2 a z + 5 a z - 3 a z - ---- + 2 a z + 7 a z - 3 a z - z z a a 5 z 5 3 5 5 5 7 3 7 > -- + 3 a z + 4 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 373]][a, z]
Out[9]=	3 2 2 a a 2 z 3 7 2 z 2 2 4 2 -a + - + -- + --- - a z - 5 a z + 2 a z + 2 z - -- + 6 a z + 8 a z + z z a 2 a 3 3 6 2 8 2 z 7 z 3 3 3 5 3 7 3 > 3 a z - 2 a z + -- - ---- - 7 a z + 9 a z + 2 a z - 6 a z - 3 a a 4 5 5 4 5 z 2 4 4 4 6 4 8 4 z 13 z > 6 z + ---- - 22 a z - 23 a z - 9 a z + 3 a z - -- + ----- + 2 3 a a a 6 5 3 5 5 5 7 5 6 4 z 2 6 > 11 a z - 16 a z - 5 a z + 8 a z + 13 z - ---- + 28 a z + 2 a 7 4 6 6 6 8 6 8 z 7 3 7 5 7 7 7 > 24 a z + 12 a z - a z - ---- + a z + 21 a z + 9 a z - 3 a z - a 8 2 8 4 8 6 8 9 3 9 5 9 > 9 z - 11 a z - 7 a z - 5 a z - 6 a z - 11 a z - 5 a z - 2 10 4 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 2 8 5 9 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 12 10 10 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a373

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