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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 11q-15/2 + 17q-13/2 - 23q-11/2 + 26q-9/2 - 26q-7/2 + 22q-5/2 - 16q-3/2 + 9q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 2q-26 + 4q-24 + 2q-22 - q-20 + 7q-18 - 3q-16 + 3q-14 - q-12 - 3q-10 + 4q-8 - 5q-6 + 4q-4 - 2 + 2q2 - q4

HOMFLY-PT Polynomial:

az + 2az3 + az5 + a3z-1 - 2a3z - 4a3z3 - 3a3z5 - a3z7 - 3a5z-1 - 6a5z - 5a5z3 - 3a5z5 - a5z7 + 2a7z-1 + 3a7z + 2a7z3 + a7z5

Kauffman Polynomial:

- z2 + 2z4 - z6 + az - 6az3 + 9az5 - 4az7 - a2 + 2a2z2 - 8a2z4 + 14a2z6 - 7a2z8 + a3z-1 + a3z - 7a3z3 + 5a3z5 + 8a3z7 - 7a3z9 - 3a4 + 17a4z2 - 36a4z4 + 39a4z6 - 12a4z8 - 3a4z10 + 3a5z-1 - 12a5z + 16a5z3 - 20a5z5 + 31a5z7 - 17a5z9 - 3a6 + 15a6z2 - 39a6z4 + 48a6z6 - 18a6z8 - 3a6z10 + 2a7z-1 - 6a7z + 3a7z3 + a7z5 + 9a7z7 - 10a7z9 + a8z2 - 10a8z4 + 20a8z6 - 13a8z8 + 6a9z - 13a9z3 + 16a9z5 - 10a9z7 + 3a10z4 - 4a10z6 + a11z3 - 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                 10 3     j = -4               13 7       j = -6             13 9         j = -8           13 13           j = -10         10 13             j = -12       7 13               j = -14     4 10                 j = -16     7                   j = -18 1 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, 335]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 335]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 3, 15, 4], X[16, 7, 17, 8], X[8, 9, 1, 10], > X[2, 15, 3, 16], X[22, 18, 9, 17], X[18, 12, 19, 11], X[4, 19, 5, 20], > X[20, 5, 21, 6], X[6, 13, 7, 14], X[12, 22, 13, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 2, -8, 9, -10, 3, -4}, > {4, -1, 7, -11, 10, -2, 5, -3, 6, -7, 8, -9, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 4 11 17 23 26 26 22 16 -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 2 4 2 -20 7 3 3 -12 3 4 5 -2 + q - --- + --- + --- - q + --- - --- + --- - q - --- + -- - -- + 26 24 22 18 16 14 10 8 6 q q q q q q q q q 4 2 4 > -- + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 335]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 3 -- - ---- + ---- + a z - 2 a z - 6 a z + 3 a z + 2 a z - 4 a z - z z z 5 3 7 3 5 3 5 5 5 7 5 3 7 5 7 > 5 a z + 2 a z + a z - 3 a z - 3 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 335]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- + a z + a z - 12 a z - 6 a z + 6 a z - z z z 2 2 2 4 2 6 2 8 2 3 3 3 5 3 > z + 2 a z + 17 a z + 15 a z + a z - 6 a z - 7 a z + 16 a z + 7 3 9 3 11 3 4 2 4 4 4 6 4 > 3 a z - 13 a z + a z + 2 z - 8 a z - 36 a z - 39 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 > 10 a z + 3 a z + 9 a z + 5 a z - 20 a z + a z + 16 a z - 11 5 6 2 6 4 6 6 6 8 6 10 6 > a z - z + 14 a z + 39 a z + 48 a z + 20 a z - 4 a z - 7 3 7 5 7 7 7 9 7 2 8 4 8 > 4 a z + 8 a z + 31 a z + 9 a z - 10 a z - 7 a z - 12 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 18 a z - 13 a z - 7 a z - 17 a z - 10 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 10 1 1 4 7 4 10 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 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 10 13 13 13 13 9 13 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 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 L11a335

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