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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 10q-4 + 14q-3 - 14q-2 + 15q-1 - 11 + 9q - 5q2 + 2q3 - q4

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-2z-2 - 4a-2 - 4a-2z2 - a-2z4 + 4z-2 + 15 + 18z2 + 10z4 + 2z6 - 5a2z-2 - 16a2 - 21a2z2 - 15a2z4 - 6a2z6 - a2z8 + 2a4z-2 + 5a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

a-3z-1 - 5a-3z + 8a-3z3 - 5a-3z5 + a-3z7 - a-2z-2 + 4a-2 - 7a-2z2 + 10a-2z4 - 8a-2z6 + 2a-2z8 + 5a-1z-1 - 21a-1z + 28a-1z3 - 12a-1z5 - 3a-1z7 + 2a-1z9 - 4z-2 + 17 - 30z2 + 37z4 - 25z6 + 4z8 + z10 + 9az-1 - 33az + 39az3 - 11az5 - 12az7 + 6az9 - 5a2z-2 + 20a2 - 39a2z2 + 47a2z4 - 35a2z6 + 9a2z8 + a2z10 + 5a3z-1 - 16a3z + 23a3z3 - 15a3z5 - a3z7 + 4a3z9 - 2a4z-2 + 6a4 - 9a4z2 + 11a4z4 - 12a4z6 + 7a4z8 + a5z + 2a5z3 - 8a5z5 + 7a5z7 - 2a6 + 6a6z2 - 8a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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 = 9                       1 j = 7                     1   j = 5                   4 1   j = 3                 5 1     j = 1               6 4       j = -1             9 5         j = -3           7 8           j = -5         7 7             j = -7       4 8               j = -9     3 6                 j = -11     4                   j = -13 1 3                     j = -15 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, 503]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 503]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > 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], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -6}, > {11, -4, 7, -9, 6, -5, 8, -7, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 10 14 14 15 2 3 4 -11 + q - -- + -- - -- + -- - -- + -- + 9 q - 5 q + 2 q - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 3 -14 3 6 2 8 2 5 2 4 6 3 + q - q + --- + q + --- + --- + -- + -- + -- + -- - q + q - 3 q - 16 12 10 8 6 4 2 q q q q q q q 8 10 12 > q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 503]][a, z]
Out[8]=	2 4 2 4 2 4 4 1 5 a 2 a 2 4 z 2 2 15 - -- - 16 a + 5 a + -- - ----- - ---- + ---- + 18 z - ---- - 21 a z + 2 2 2 2 2 2 2 a z a z z z a 4 4 2 4 z 2 4 4 4 6 2 6 4 6 2 8 > 6 a z + 10 z - -- - 15 a z + 4 a z + 2 z - 6 a z + a z - a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 503]][a, z]
Out[9]=	2 4 4 2 4 6 4 1 5 a 2 a 1 5 9 a 17 + -- + 20 a + 6 a - 2 a - -- - ----- - ---- - ---- + ---- + --- + --- + 2 2 2 2 2 2 3 a z z a z a z z z a z 3 2 5 a 5 z 21 z 3 5 2 7 z 2 2 > ---- - --- - ---- - 33 a z - 16 a z + a z - 30 z - ---- - 39 a z - z 3 a 2 a a 3 3 4 2 6 2 8 2 8 z 28 z 3 3 3 5 3 > 9 a z + 6 a z - a z + ---- + ----- + 39 a z + 23 a z + 2 a z - 3 a a 4 5 7 3 4 10 z 2 4 4 4 6 4 8 4 5 z > 2 a z + 37 z + ----- + 47 a z + 11 a z - 8 a z + a z - ---- - 2 3 a a 5 6 12 z 5 3 5 5 5 7 5 6 8 z 2 6 > ----- - 11 a z - 15 a z - 8 a z + 3 a z - 25 z - ---- - 35 a z - a 2 a 7 7 8 4 6 6 6 z 3 z 7 3 7 5 7 8 2 z > 12 a z + 6 a z + -- - ---- - 12 a z - a z + 7 a z + 4 z + ---- + 3 a 2 a a 9 2 8 4 8 2 z 9 3 9 10 2 10 > 9 a z + 7 a z + ---- + 6 a z + 4 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	8 9 1 1 3 4 3 6 4 8 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 7 7 5 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 4 q t + 5 q t + q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a503

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