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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 9q-5/2 - 17q-3/2 + 22q-1/2 - 26q1/2 + 26q3/2 - 23q5/2 + 17q7/2 - 10q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 3q-8 + 6q-6 + q-2 + 6 - 4q2 + 4q4 - 5q6 + 2q10 - 4q12 + 5q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 3a-3z-1 + 5a-3z + 4a-3z3 + 2a-3z5 - 4a-1z-1 - 7a-1z - 6a-1z3 - 3a-1z5 - a-1z7 + 2az-1 + 4az + 4az3 + 2az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 4a-6z4 - 4a-6z6 - a-5z-1 + 4a-5z - 10a-5z3 + 13a-5z5 - 9a-5z7 - a-4 + 5a-4z2 - 13a-4z4 + 18a-4z6 - 12a-4z8 - 3a-3z-1 + 16a-3z - 34a-3z3 + 32a-3z5 - 4a-3z7 - 8a-3z9 - 3a-2 + 17a-2z2 - 45a-2z4 + 58a-2z6 - 24a-2z8 - 2a-2z10 - 4a-1z-1 + 22a-1z - 45a-1z3 + 36a-1z5 + 8a-1z7 - 14a-1z9 - 2 + 13z2 - 37z4 + 50z6 - 19z8 - 2z10 - 2az-1 + 13az - 30az3 + 27az5 - az7 - 6az9 - a2 + a2z2 - 7a2z4 + 13a2z6 - 7a2z8 + 3a3z - 8a3z3 + 9a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   7 1   j = 8                 10 3     j = 6               13 7       j = 4             13 10         j = 2           13 13           j = 0         11 15             j = -2       6 11               j = -4     3 11                 j = -6   1 6                   j = -8   3                     j = -10 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, 28]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 28]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 15, 7], X[22, 16, 5, 15], > X[16, 9, 17, 10], X[8, 21, 9, 22], X[20, 17, 21, 18], X[18, 14, 19, 13], > X[12, 20, 13, 19], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -9, 8, -3, 4, -5, 7, -8, > 9, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 9 17 22 3/2 5/2 q - ---- + ---- - ---- + ------- - 26 Sqrt[q] + 26 q - 23 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 17 q - 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 3 6 -2 2 4 6 10 12 6 - q + q + q - -- + -- + q - 4 q + 4 q - 5 q + 2 q - 4 q + 8 6 q q 14 18 20 > 5 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 28]][a, z]
Out[8]=	3 3 3 1 3 4 2 a z 5 z 7 z 3 z 4 z 6 z -(----) + ---- - --- + --- - -- + --- - --- + 4 a z - a z - -- + ---- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 7 3 3 3 2 z 3 z 5 z > 4 a z - a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 28]][a, z]
Out[9]=	-4 3 2 1 3 4 2 a 4 z 16 z 22 z -2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 13 a z + 2 5 3 a z z 5 3 a a a z a z a a 2 2 2 3 3 3 3 2 z 5 z 17 z 2 2 4 2 z 10 z 34 z > 3 a z + 13 z - -- + ---- + ----- + a z - a z + -- - ----- - ----- - 6 4 2 7 5 3 a a a a a a 3 4 4 4 45 z 3 3 3 4 4 z 13 z 45 z 2 4 > ----- - 30 a z - 8 a z - 37 z + ---- - ----- - ----- - 7 a z + a 6 4 2 a a a 5 5 5 5 6 4 4 z 13 z 32 z 36 z 5 3 5 6 4 z > 2 a z - -- + ----- + ----- + ----- + 27 a z + 9 a z + 50 z - ---- + 7 5 3 a 6 a a a a 6 6 7 7 7 18 z 58 z 2 6 4 6 9 z 4 z 8 z 7 3 7 > ----- + ----- + 13 a z - a z - ---- - ---- + ---- - a z - 4 a z - 4 2 5 3 a a a a a 8 8 9 9 10 8 12 z 24 z 2 8 8 z 14 z 9 10 2 z > 19 z - ----- - ----- - 7 a z - ---- - ----- - 6 a z - 2 z - ----- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 11 6 11 15 + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 11 2 4 4 2 6 2 6 3 8 3 > ---- + 13 q t + 13 q t + 10 q t + 13 q t + 7 q t + 10 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a28

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