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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 7q-4 - 11q-3 + 19q-2 - 20q-1 + 22 - 20q + 15q2 - 10q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 5q-14 + 7q-12 + 2q-10 + 10q-8 + 5q-6 + 2q-4 + 5q-2 - 4 + 2q2 - 6q4 - 3q6 + q8 - 5q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 - a-2z-2 - 3a-2 + 2a-2z4 + 3z-2 + 7 + 4z2 - z6 - 2a2z-2 - a2 + 3a2z2 + 3a2z4 - a4z-2 - 4a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 4a-4z4 + 4a-4z6 + 2a-3z-1 - 7a-3z + 12a-3z3 - 15a-3z5 + 9a-3z7 - a-2z-2 + a-2 - 2a-2z2 + 4a-2z4 - 13a-2z6 + 10a-2z8 + 8a-1z-1 - 27a-1z + 50a-1z3 - 47a-1z5 + 12a-1z7 + 5a-1z9 - 3z-2 + 5 - 14z2 + 34z4 - 43z6 + 19z8 + z10 + 10az-1 - 34az + 53az3 - 39az5 + 2az7 + 8az9 - 2a2z-2 + 4a2 - 11a2z2 + 26a2z4 - 30a2z6 + 12a2z8 + a2z10 + 2a3z-1 - 10a3z + 16a3z3 - 12a3z5 + a3z7 + 3a3z9 + a4z-2 - 3a4 + 6a4z2 - 4a4z4 - 3a4z6 + 3a4z8 - 2a5z-1 + 4a5z - 4a5z5 + 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                   7 1   j = 5                 8 3     j = 3               12 7       j = 1             10 8         j = -1           11 13           j = -3         8 9             j = -5       3 11               j = -7     4 8                 j = -9   1 6                   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, 386]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 386]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[18, 12, 19, 11], X[22, 20, 9, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, 5, -9, 4, -3, 7, -8, 9, -5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 7 11 19 20 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 20 q + 15 q - 10 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 5 7 2 10 5 2 5 2 4 -4 + q + --- + q + --- + --- + --- + -- + -- + -- + -- + 2 q - 6 q - 18 14 12 10 8 6 4 2 q q q q q q q q 6 8 10 12 14 16 > 3 q + q - 5 q + 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 386]][a, z]
Out[8]=	2 4 6 2 3 2 4 6 3 1 2 a a a 2 z 2 2 7 - -- - a - 4 a + a + -- - ----- - ---- - -- + -- + 4 z - -- + 3 a z - 2 2 2 2 2 2 2 4 a z a z z z z a 4 4 2 2 z 2 4 6 > 3 a z + ---- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 386]][a, z]
Out[9]=	2 4 6 -2 2 4 6 3 1 2 a a a 2 8 5 + a + 4 a - 3 a - 4 a - -- - ----- - ---- + -- + -- + ---- + --- + 2 2 2 2 2 2 3 a z z a z z z z a z 3 5 2 10 a 2 a 2 a 7 z 27 z 3 5 2 z > ---- + ---- - ---- - --- - ---- - 34 a z - 10 a z + 4 a z - 14 z + -- - z z z 3 a 4 a a 2 3 3 3 2 z 2 2 4 2 6 2 z 12 z 50 z 3 > ---- - 11 a z + 6 a z + 6 a z - -- + ----- + ----- + 53 a z + 2 5 3 a a a a 4 4 5 3 3 4 4 z 4 z 2 4 4 4 6 4 z > 16 a z + 34 z - ---- + ---- + 26 a z - 4 a z - 4 a z + -- - 4 2 5 a a a 5 5 6 6 15 z 47 z 5 3 5 5 5 6 4 z 13 z > ----- - ----- - 39 a z - 12 a z - 4 a z - 43 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 9 z 12 z 7 3 7 5 7 > 30 a z - 3 a z + a z + ---- + ----- + 2 a z + a z + 2 a z + 3 a a 8 9 8 10 z 2 8 4 8 5 z 9 3 9 10 2 10 > 19 z + ----- + 12 a z + 3 a z + ---- + 8 a z + 3 a z + z + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 1 1 6 4 8 3 11 -- + 10 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 8 9 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 12 q t + 7 q t + 8 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a386

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