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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 3q-15/2 - 4q-13/2 + 5q-11/2 - 6q-9/2 + 5q-7/2 - 5q-5/2 + 4q-3/2 - 3q-1/2 + q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-28 - q-18 + q-16 + q-12 + q-10 + q-6 + q-4 + 2q-2 + 2 + q2 + q4

HOMFLY-PT Polynomial:

- 2az-1 - 7az - 5az3 - az5 + 3a3z-1 + 10a3z + 12a3z3 + 6a3z5 + a3z7 - a5z-1 + 6a5z3 + 5a5z5 + a5z7 - 3a7z - 4a7z3 - a7z5

Kauffman Polynomial:

2az-1 - 13az + 28az3 - 23az5 + 8az7 - az9 - 3a2 + 7a2z2 + a2z4 - 10a2z6 + 6a2z8 - a2z10 + 3a3z-1 - 19a3z + 51a3z3 - 57a3z5 + 26a3z7 - 4a3z9 - 3a4 + 21a4z2 - 37a4z4 + 16a4z6 + a4z8 - a4z10 + a5z-1 - 5a5z3 - 8a5z5 + 12a5z7 - 3a5z9 - a6 + 8a6z2 - 23a6z4 + 21a6z6 - 5a6z8 + 6a7z - 20a7z3 + 22a7z5 - 6a7z7 - 3a8z2 + 12a8z4 - 5a8z6 + 6a9z3 - 4a9z5 + 2a10z2 - 3a10z4 - 2a11z3 - a12z2

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 = 4                       1 j = 2                         j = 0                   3 1   j = -2                 1       j = -4               4 3       j = -6             3 3         j = -8           3 2           j = -10         2 3             j = -12       2 3               j = -14     1 2                 j = -16   1 2                   j = -18   1                     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, 318]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 318]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], > X[20, 13, 21, 14], X[16, 8, 17, 7], X[18, 6, 19, 5], X[6, 18, 7, 17], > X[4, 20, 5, 19], X[22, 15, 9, 16], X[14, 21, 15, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 7, -8, 6, -4}, > {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 4 5 6 5 5 4 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 3 3/2 > ------- + Sqrt[q] - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -18 -16 -12 -10 -6 -4 2 2 4 2 - q - q + q + q + q + q + q + -- + q + q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 318]][a, z]
Out[8]=	3 5 -2 a 3 a a 3 7 3 3 3 5 3 ---- + ---- - -- - 7 a z + 10 a z - 3 a z - 5 a z + 12 a z + 6 a z - z z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 4 a z - a z + 6 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 318]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 7 2 2 -3 a - 3 a - a + --- + ---- + -- - 13 a z - 19 a z + 6 a z + 7 a z + z z z 4 2 6 2 8 2 10 2 12 2 3 3 3 > 21 a z + 8 a z - 3 a z + 2 a z - a z + 28 a z + 51 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 5 a z - 20 a z + 6 a z - 2 a z + a z - 37 a z - 23 a z + 8 4 10 4 5 3 5 5 5 7 5 9 5 > 12 a z - 3 a z - 23 a z - 57 a z - 8 a z + 22 a z - 4 a z - 2 6 4 6 6 6 8 6 7 3 7 5 7 > 10 a z + 16 a z + 21 a z - 5 a z + 8 a z + 26 a z + 12 a z - 7 7 2 8 4 8 6 8 9 3 9 5 9 2 10 > 6 a z + 6 a z + a z - 5 a z - a z - 4 a z - 3 a z - a z - 4 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 2 1 2 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 3 2 3 3 2 3 3 t t 2 3 > ------ + ------ + ------ + ----- + ---- + ---- + --- + -- + 3 t + t + 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q 4 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a318

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