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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X6718 X22,15,7,16 X4,14,5,13 X16,6,17,5 X12,18,13,17 X10,4,11,3 X18,12,19,11 X14,21,15,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 15q-5/2 + 20q-3/2 - 23q-1/2 + 22q1/2 - 20q3/2 + 14q5/2 - 9q7/2 + 4q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 2q-12 + 3q-8 - 4q-6 + 4q-4 - 2q-2 + 1 + 3q2 - 2q4 + 6q6 - q8 + q10 + 2q12 - 2q14 + q16

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - 2a-3z3 - a-3z5 + a-1z-1 + 2a-1z + 3a-1z3 + 3a-1z5 + a-1z7 + az + 3az3 + 3az5 + az7 - a3z - 2a3z3 - a3z5

Kauffman Polynomial:

a-5z - 3a-5z3 + 3a-5z5 - a-5z7 + 4a-4z2 - 13a-4z4 + 13a-4z6 - 4a-4z8 + a-3z-1 - a-3z3 - 11a-3z5 + 17a-3z7 - 6a-3z9 - a-2 + 10a-2z2 - 38a-2z4 + 37a-2z6 - 5a-2z8 - 3a-2z10 + a-1z-1 - 2a-1z3 - 20a-1z5 + 40a-1z7 - 16a-1z9 + 10z2 - 44z4 + 57z6 - 16z8 - 3z10 + 3az - 15az3 + 17az5 + 8az7 - 10az9 + 2a2z2 - 11a2z4 + 24a2z6 - 15a2z8 + 2a3z - 10a3z3 + 19a3z5 - 14a3z7 - 2a4z2 + 7a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

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 = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 8 3     j = 4               12 6       j = 2             11 9         j = 0           12 11           j = -2         9 12             j = -4       6 11               j = -6     3 9                 j = -8   1 6                   j = -10   3                     j = -12 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, 210]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 210]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[6, 7, 1, 8], X[22, 15, 7, 16], > X[4, 14, 5, 13], X[16, 6, 17, 5], X[12, 18, 13, 17], X[10, 4, 11, 3], > X[18, 12, 19, 11], X[14, 21, 15, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -5, 6, -3}, > {3, -1, 2, -8, 9, -7, 5, -10, 4, -6, 7, -9, 11, -2, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 15 20 23 3/2 q - ---- + ---- - ---- + ---- - ------- + 22 Sqrt[q] - 20 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 14 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 3 4 4 2 2 4 6 8 10 1 - q + --- - --- + -- - -- + -- - -- + 3 q - 2 q + 6 q - q + q + 14 12 8 6 4 2 q q q q q q 12 14 16 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 210]][a, z]
Out[8]=	3 3 5 1 1 z 2 z 3 2 z 3 z 3 3 3 z -(----) + --- - -- + --- + a z - a z - ---- + ---- + 3 a z - 2 a z - -- + 3 a z 3 a 3 a 3 a z a a a 5 7 3 z 5 3 5 z 7 > ---- + 3 a z - a z + -- + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 210]][a, z]
Out[9]=	2 2 -2 1 1 z 3 2 4 z 10 z 2 2 -a + ---- + --- + -- + 3 a z + 2 a z + 10 z + ---- + ----- + 2 a z - 3 a z 5 4 2 a z a a a 3 3 3 4 4 2 3 z z 2 z 3 3 3 5 3 4 13 z > 2 a z - ---- - -- - ---- - 15 a z - 10 a z + a z - 44 z - ----- - 5 3 a 4 a a a 4 5 5 5 38 z 2 4 4 4 6 4 3 z 11 z 20 z 5 > ----- - 11 a z + 7 a z - a z + ---- - ----- - ----- + 17 a z + 2 5 3 a a a a 6 6 7 3 5 5 5 6 13 z 37 z 2 6 4 6 z > 19 a z - 4 a z + 57 z + ----- + ----- + 24 a z - 9 a z - -- + 4 2 5 a a a 7 7 8 8 9 17 z 40 z 7 3 7 8 4 z 5 z 2 8 6 z > ----- + ----- + 8 a z - 14 a z - 16 z - ---- - ---- - 15 a z - ---- - 3 a 4 2 3 a a a a 9 10 16 z 9 10 3 z > ----- - 10 a z - 3 z - ----- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 9 6 11 12 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 9 2 2 2 4 2 4 3 6 3 6 4 > ---- + 11 t + 11 q t + 9 q t + 12 q t + 6 q t + 8 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a210

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