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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 15q-7/2 - 20q-5/2 + 23q-3/2 - 24q-1/2 + 19q1/2 - 15q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - 2q-16 + 3q-14 - 3q-12 + q-10 + q-8 - 4q-6 + 6q-4 - 3q-2 + 7 + q2 + 3q6 - 3q8 + 2q10 - q12

HOMFLY-PT Polynomial:

- a-1z-1 + 2a-1z + 5a-1z3 + 4a-1z5 + a-1z7 + az-1 - 4az - 12az3 - 13az5 - 6az7 - az9 + 2a3z + 5a3z3 + 4a3z5 + a3z7

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 5a-3z3 + 9a-3z5 - 4a-3z7 + a-2z2 - 7a-2z4 + 15a-2z6 - 7a-2z8 - a-1z-1 - 3a-1z5 + 12a-1z7 - 7a-1z9 + 1 + 6z2 - 22z4 + 27z6 - 7z8 - 3z10 - az-1 - 2az + 10az3 - 27az5 + 33az7 - 15az9 + 8a2z2 - 28a2z4 + 31a2z6 - 10a2z8 - 3a2z10 - a3z3 - 2a3z5 + 9a3z7 - 8a3z9 + 3a4z2 - 10a4z4 + 16a4z6 - 10a4z8 + a5z - 5a5z3 + 12a5z5 - 8a5z7 - a6z2 + 5a6z4 - 4a6z6 + a7z3 - a7z5

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 = 10                       1 j = 8                     3   j = 6                   6 1   j = 4                 9 3     j = 2               10 6       j = 0             14 9         j = -2           11 12           j = -4         9 12             j = -6       6 11               j = -8     3 9                 j = -10   1 6                   j = -12   3                     j = -14 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, 326]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 326]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 5, 21, 6], X[16, 9, 17, 10], > X[18, 12, 19, 11], X[22, 15, 9, 16], X[6, 21, 7, 22], X[14, 8, 15, 7], > X[4, 14, 5, 13], X[8, 18, 1, 17], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 3, -7, 8, -10}, > {4, -1, 5, -2, 9, -8, 6, -4, 10, -5, 11, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 15 20 23 24 -q + ----- - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 3 3 -10 -8 4 6 3 2 6 8 7 + q - --- + --- - --- + q + q - -- + -- - -- + q + 3 q - 3 q + 16 14 12 6 4 2 q q q q q q 10 12 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 326]][a, z]
Out[8]=	3 5 1 a 2 z 3 5 z 3 3 3 4 z 5 -(---) + - + --- - 4 a z + 2 a z + ---- - 12 a z + 5 a z + ---- - 13 a z + a z z a a a 7 3 5 z 7 3 7 9 > 4 a z + -- - 6 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 326]][a, z]
Out[9]=	2 2 1 a z 5 2 z z 2 2 4 2 6 2 1 - --- - - + -- - 2 a z + a z + 6 z - -- + -- + 8 a z + 3 a z - a z - a z z 3 4 2 a a a 3 4 4 5 z 3 3 3 5 3 7 3 4 2 z 7 z 2 4 > ---- + 10 a z - a z - 5 a z + a z - 22 z + ---- - ---- - 28 a z - 3 4 2 a a a 5 5 4 4 6 4 9 z 3 z 5 3 5 5 5 7 5 > 10 a z + 5 a z + ---- - ---- - 27 a z - 2 a z + 12 a z - a z + 3 a a 6 6 7 7 6 z 15 z 2 6 4 6 6 6 4 z 12 z > 27 z - -- + ----- + 31 a z + 16 a z - 4 a z - ---- + ----- + 4 2 3 a a a a 8 9 7 3 7 5 7 8 7 z 2 8 4 8 7 z > 33 a z + 9 a z - 8 a z - 7 z - ---- - 10 a z - 10 a z - ---- - 2 a a 9 3 9 10 2 10 > 15 a z - 8 a z - 3 z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 9 6 11 14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 9 12 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 10 q t + 6 q t + 9 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a326

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