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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 5q-17/2 - 11q-15/2 + 18q-13/2 - 24q-11/2 + 27q-9/2 - 27q-7/2 + 22q-5/2 - 17q-3/2 + 9q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 3q-26 + 2q-24 - q-22 - 4q-20 + 5q-18 - 4q-16 + 4q-14 + q-12 + 7q-8 - 3q-6 + 5q-4 - 2 + 2q2 - q4

HOMFLY-PT Polynomial:

az + 2az3 + az5 - 2a3z-1 - 6a3z - 5a3z3 - 3a3z5 - a3z7 + 3a5z-1 + 5a5z + a5z3 - 2a5z5 - a5z7 - a7z-1 - a7z + a7z3 + a7z5

Kauffman Polynomial:

- z2 + 2z4 - z6 + 2az - 7az3 + 9az5 - 4az7 - 6a2z4 + 13a2z6 - 7a2z8 - 2a3z-1 + 9a3z - 16a3z3 + 11a3z5 + 6a3z7 - 7a3z9 + 3a4 - 23a4z4 + 37a4z6 - 13a4z8 - 3a4z10 - 3a5z-1 + 10a5z - 17a5z3 + 4a5z5 + 24a5z7 - 17a5z9 + 3a6 - 3a6z2 - 24a6z4 + 47a6z6 - 20a6z8 - 3a6z10 - a7z-1 + 3a7z - 12a7z3 + 17a7z5 + 3a7z7 - 10a7z9 + a8 - 2a8z2 - 5a8z4 + 19a8z6 - 14a8z8 - 4a9z3 + 14a9z5 - 11a9z7 + 4a10z4 - 5a10z6 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   6 1   j = -2                 11 3     j = -4               12 7       j = -6             15 10         j = -8           13 13           j = -10         11 14             j = -12       7 13               j = -14     4 11                 j = -16   1 7                   j = -18   4                     j = -20 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, 143]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 143]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[22, 19, 7, 20], > X[12, 5, 13, 6], X[10, 4, 11, 3], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[20, 13, 21, 14], X[14, 21, 15, 22], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 6, -7, 5, -3}, > {3, -1, 2, -6, 8, -5, 9, -10, 7, -8, 11, -2, 4, -9, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 5 11 18 24 27 27 22 17 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 9 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 2 -22 4 5 4 4 -12 7 3 5 -2 + q - --- + --- - q - --- + --- - --- + --- + q + -- - -- + -- + 26 24 20 18 16 14 8 6 4 q q q q q q q q q 2 4 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 143]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 3 5 3 ----- + ---- - -- + a z - 6 a z + 5 a z - a z + 2 a z - 5 a z + a z + z z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > a z + a z - 3 a z - 2 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 143]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 2 3 a + 3 a + a - ---- - ---- - -- + 2 a z + 9 a z + 10 a z + 3 a z - z - z z z 6 2 8 2 3 3 3 5 3 7 3 9 3 > 3 a z - 2 a z - 7 a z - 16 a z - 17 a z - 12 a z - 4 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > 2 z - 6 a z - 23 a z - 24 a z - 5 a z + 4 a z + 9 a z + 3 5 5 5 7 5 9 5 11 5 6 2 6 > 11 a z + 4 a z + 17 a z + 14 a z - a z - z + 13 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 37 a z + 47 a z + 19 a z - 5 a z - 4 a z + 6 a z + 24 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 3 a z - 11 a z - 7 a z - 13 a z - 20 a z - 14 a z - 7 a z - 5 9 7 9 4 10 6 10 > 17 a z - 10 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 11 1 4 1 7 4 11 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 13 11 14 13 13 15 10 12 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a143

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