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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X14,12,15,11 X12,5,13,6 X22,13,5,14 X10,19,11,20 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 10q-11/2 - 17q-9/2 + 23q-7/2 - 27q-5/2 + 27q-3/2 - 25q-1/2 + 17q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 3q-18 - q-16 + 2q-14 - 6q-12 + 4q-10 + 2q-6 + 6q-4 - 2q-2 + 8 - 3q2 - q4 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 - 2az-1 - az + 4az3 + 3az5 + az7 + 3a3z-1 - 5a3z3 - 3a3z5 - a5z-1 + 2a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - 7a-1z3 + 15a-1z5 - 11a-1z7 + 2z2 - 4z4 + 15z6 - 13z8 + 2az-1 - az - 25az3 + 41az5 - 10az7 - 8az9 - 3a2 + 11a2z2 - 39a2z4 + 61a2z6 - 27a2z8 - 2a2z10 + 3a3z-1 + a3z - 30a3z3 + 35a3z5 + 8a3z7 - 15a3z9 - 3a4 + 16a4z2 - 47a4z4 + 59a4z6 - 22a4z8 - 2a4z10 + a5z-1 + 4a5z - 18a5z3 + 18a5z5 + 3a5z7 - 7a5z9 - a6 + 6a6z2 - 14a6z4 + 17a6z6 - 8a6z8 + 2a7z - 6a7z3 + 8a7z5 - 4a7z7 - a8z2 + 2a8z4 - 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                     4   j = 4                   7 1   j = 2                 10 4     j = 0               15 7       j = -2             14 12         j = -4           13 13           j = -6         10 14             j = -8       7 13               j = -10     3 10                 j = -12   1 7                   j = -14   3                     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, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[20, 9, 21, 10], > X[8, 4, 9, 3], X[18, 22, 19, 21], X[14, 12, 15, 11], X[12, 5, 13, 6], > X[22, 13, 5, 14], X[10, 19, 11, 20], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, > 10, -4, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 10 17 23 27 27 25 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 17 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 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 -16 2 6 4 2 6 2 2 4 8 - q + --- - --- - q + --- - --- + --- + -- + -- - -- - 3 q - q + 20 18 14 12 10 6 4 2 q q q q q q q q 6 8 10 > 3 q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 8]][a, z]
Out[8]=	3 5 3 5 -2 a 3 a a 5 7 z 3 3 3 5 3 z ---- + ---- - -- - a z + 2 a z - a z - -- + 4 a z - 5 a z + 3 a z - -- + z z z a a 5 3 5 7 > 3 a z - 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 8]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 7 2 -3 a - 3 a - a + --- + ---- + -- - a z + a z + 4 a z + 2 a z + 2 z + z z z 3 2 2 4 2 6 2 8 2 7 z 3 3 3 > 11 a z + 16 a z + 6 a z - a z - ---- - 25 a z - 30 a z - a 4 5 3 7 3 4 4 z 2 4 4 4 6 4 > 18 a z - 6 a z - 4 z + ---- - 39 a z - 47 a z - 14 a z + 2 a 5 5 8 4 z 15 z 5 3 5 5 5 7 5 6 > 2 a z - -- + ----- + 41 a z + 35 a z + 18 a z + 8 a z + 15 z - 3 a a 6 7 5 z 2 6 4 6 6 6 8 6 11 z 7 3 7 > ---- + 61 a z + 59 a z + 17 a z - a z - ----- - 10 a z + 8 a z + 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 > 3 a z - 4 a z - 13 z - 27 a z - 22 a z - 8 a z - 8 a z - 3 9 5 9 2 10 4 10 > 15 a z - 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 7 3 10 7 13 15 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 10 14 13 13 14 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 7 t + 10 q t + 4 q t + 7 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 + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a8

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