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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X20,12,21,11 X22,17,9,18 X18,21,19,22 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,20,13,19 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 6q-4 - 10q-3 + 16q-2 - 18q-1 + 19 - 17q + 14q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 4q-14 + 4q-12 - 2q-10 + 4q-8 - q-4 + 4q-2 - 1 + 6q2 + 2q6 + 4q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + 2a-2 + 2a-2z2 + 2a-2z4 - 3z-2 - 6 - 5z2 - 2z4 - z6 + 4a2z-2 + 8a2 + 6a2z2 + 3a2z4 - 3a4z-2 - 5a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 3a-4z2 - 6a-4z4 + 4a-4z6 + 4a-3z3 - 10a-3z5 + 7a-3z7 + a-2z-2 - 4a-2 + 8a-2z2 - 7a-2z4 - 5a-2z6 + 7a-2z8 - a-1z-1 + a-1z + 5a-1z3 - 15a-1z5 + 5a-1z7 + 4a-1z9 + 3z-2 - 14 + 24z2 - 13z4 - 11z6 + 10z8 + z10 - az-1 + az + 5az3 - 11az5 - az7 + 6az9 + 4a2z-2 - 21a2 + 38a2z2 - 26a2z4 - a2z6 + 5a2z8 + a2z10 - a3z-1 + a3z + 8a3z3 - 12a3z5 + 3a3z7 + 2a3z9 + 3a4z-2 - 14a4 + 25a4z2 - 18a4z4 + 2a4z6 + 2a4z8 - a5z-1 + a5z + 3a5z3 - 5a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 4a6z4 + 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                     3   j = 7                   5 1   j = 5                 9 3     j = 3               8 5       j = 1             11 9         j = -1           10 11           j = -3         6 8             j = -5       4 10               j = -7     2 6                 j = -9   1 5                   j = -11   1                     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, 389]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 389]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 12, 21, 11], X[22, 17, 9, 18], > X[18, 21, 19, 22], X[16, 14, 17, 13], X[8, 16, 5, 15], X[14, 8, 15, 7], > X[12, 20, 13, 19], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -7}, > {11, -2, 3, -9, 6, -8, 7, -6, 4, -5, 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 6 10 16 18 2 3 4 5 19 + q - -- + -- - -- + -- - -- - 17 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 4 2 4 -4 4 2 6 8 -1 + q + --- + q + --- + --- - --- + -- - q + -- + 6 q + 2 q + 4 q - 18 14 12 10 8 2 q q q q q q 10 12 14 16 > 3 q + 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 389]][a, z]
Out[8]=	2 4 6 2 2 2 2 4 6 3 1 4 a 3 a a 2 z 2 z -6 + -- + 8 a - 5 a + a - -- + ----- + ---- - ---- + -- - 5 z - -- + ---- + 2 2 2 2 2 2 2 4 2 a z a z z z z a a 4 2 2 4 2 4 2 z 2 4 6 > 6 a z - 3 a z - 2 z + ---- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 389]][a, z]
Out[9]=	2 4 6 4 2 4 6 3 1 4 a 3 a a 1 a -14 - -- - 21 a - 14 a - 4 a + -- + ----- + ---- + ---- + -- - --- - - - 2 2 2 2 2 2 2 a z z a z a z z z z 3 5 2 2 a a z 3 5 2 3 z 8 z 2 2 > -- - -- + - + a z + a z + a z + 24 z + ---- + ---- + 38 a z + z z a 4 2 a a 3 3 3 4 2 6 2 z 4 z 5 z 3 3 3 5 3 > 25 a z + 6 a z - -- + ---- + ---- + 5 a z + 8 a z + 3 a z - 5 3 a a a 4 4 5 5 5 4 6 z 7 z 2 4 4 4 6 4 z 10 z 15 z > 13 z - ---- - ---- - 26 a z - 18 a z - 4 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 4 z 5 z 2 6 4 6 > 11 a z - 12 a z - 5 a z - 11 z + ---- - ---- - a z + 2 a z + 4 2 a a 7 7 8 6 6 7 z 5 z 7 3 7 5 7 8 7 z 2 8 > a z + ---- + ---- - a z + 3 a z + 2 a z + 10 z + ---- + 5 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 2 a z + ---- + 6 a z + 2 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 1 1 5 2 6 4 10 -- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 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 6 8 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 8 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a389

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