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DrawMorseLink

PD Presentation:

X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 11q-3 + 17q-2 - 18q-1 + 19 - 16q + 12q2 - 7q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-14 + 5q-12 + 2q-10 + 10q-8 + 5q-6 + 4q-4 + 4q-2 - 3 + 2q2 - 4q4 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 - a-2z-2 + a-2 + 3a-2z2 + 2a-2z4 + 4z-2 + 6 + 2z2 - z4 - z6 - 5a2z-2 - 9a2 - 7a2z2 - 3a2z4 - a2z6 + 2a4z-2 + 3a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + a-3z-1 - 2a-3z + 4a-3z3 - 7a-3z5 + 5a-3z7 - a-2z-2 + 5a-2z2 - 6a-2z4 - 2a-2z6 + 5a-2z8 + 5a-1z-1 - 19a-1z + 31a-1z3 - 27a-1z5 + 8a-1z7 + 3a-1z9 - 4z-2 + 13 - 16z2 + 21z4 - 24z6 + 11z8 + z10 + 9az-1 - 35az + 50az3 - 29az5 - 2az7 + 7az9 - 5a2z-2 + 22a2 - 43a2z2 + 51a2z4 - 39a2z6 + 12a2z8 + a2z10 + 5a3z-1 - 19a3z + 28a3z3 - 17a3z5 - 2a3z7 + 4a3z9 - 2a4z-2 + 11a4 - 23a4z2 + 26a4z4 - 19a4z6 + 6a4z8 + 3a5z3 - 7a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   5 1   j = 5                 7 2     j = 3               9 5       j = 1             10 7         j = -1           10 11           j = -3         7 8             j = -5       5 11               j = -7     3 6                 j = -9     5                   j = -11 1 3                     j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 501]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 501]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 5, 15, 6], X[10, 3, 11, 4], X[4, 13, 5, 14], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[18, 12, 19, 11], X[12, 18, 7, 17], > X[20, 16, 21, 15], X[22, 20, 13, 19], X[16, 22, 17, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, > {4, -2, 9, -11, 8, -7, 10, -9, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 8 11 17 18 2 3 4 5 19 + q - -- + -- - -- + -- - -- - 16 q + 12 q - 7 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 5 2 10 5 4 4 2 4 8 -3 + q - q + --- + --- + --- + -- + -- + -- + -- + 2 q - 4 q + 2 q - 14 12 10 8 6 4 2 q q q q q q q 10 12 16 > 3 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 501]][a, z]
Out[8]=	2 4 2 2 -4 -2 2 4 4 1 5 a 2 a 2 z 3 z 6 - a + a - 9 a + 3 a + -- - ----- - ---- + ---- + 2 z - -- + ---- - 2 2 2 2 2 4 2 z a z z z a a 4 2 2 4 2 4 2 z 2 4 4 4 6 2 6 > 7 a z + 2 a z - z + ---- - 3 a z + a z - z - a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 501]][a, z]
Out[9]=	2 4 3 -4 2 4 4 1 5 a 2 a 1 5 9 a 5 a 13 - a + 22 a + 11 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 2 2 2 2 2 3 a z z z z a z z z a z 2 2 z 2 z 19 z 3 2 3 z 5 z 2 2 > -- - --- - ---- - 35 a z - 19 a z - 16 z + ---- + ---- - 43 a z - 5 3 a 4 2 a a a a 3 3 3 4 2 6 2 2 z 4 z 31 z 3 3 3 5 3 > 23 a z + 2 a z - ---- + ---- + ----- + 50 a z + 28 a z + 3 a z + 5 3 a a a 4 4 5 5 5 4 5 z 6 z 2 4 4 4 6 4 z 7 z 27 z > 21 z - ---- - ---- + 51 a z + 26 a z - 3 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 2 z 2 6 4 6 > 29 a z - 17 a z - 7 a z - 24 z + ---- - ---- - 39 a z - 19 a z + 4 2 a a 7 7 8 6 6 5 z 8 z 7 3 7 5 7 8 5 z > a z + ---- + ---- - 2 a z - 2 a z + 3 a z + 11 z + ---- + 3 a 2 a a 9 2 8 4 8 3 z 9 3 9 10 2 10 > 12 a z + 6 a z + ---- + 7 a z + 4 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 1 3 5 3 6 5 11 -- + 10 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 8 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 9 q t + 5 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a501

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