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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 6q-7 - 10q-6 + 16q-5 - 19q-4 + 21q-3 - 18q-2 + 15q-1 - 10 + 5q - q2

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 2q-26 + 2q-24 + 5q-22 - q-20 - q-18 + 4q-16 - 2q-14 + 4q-12 + 2q-10 + 2q-8 + 5q-6 - 3q-4 + 5q-2 - 1 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 + a2z-2 + 3a2 + 4a2z2 + 2a2z4 + a2z6 - 3a4z-2 - 7a4 - 7a4z2 - 4a4z4 + 4a6z-2 + 8a6 + 6a6z2 - 3a8z-2 - 4a8 + a10z-2

Kauffman Polynomial:

a-1z5 - 5z4 + 5z6 + 3az3 - 15az5 + 10az7 + a2z-2 - 4a2 + 7a2z2 - 3a2z4 - 12a2z6 + 10a2z8 - a3z-1 + a3z + 10a3z3 - 22a3z5 + 6a3z7 + 5a3z9 + 3a4z-2 - 14a4 + 23a4z2 - 3a4z4 - 23a4z6 + 14a4z8 + a4z10 - a5z-1 + a5z + 10a5z3 - 12a5z5 - 3a5z7 + 7a5z9 + 4a6z-2 - 21a6 + 35a6z2 - 19a6z4 - 5a6z6 + 6a6z8 + a6z10 - a7z-1 + a7z + 6a7z3 - 11a7z5 + 3a7z7 + 2a7z9 + 3a8z-2 - 14a8 + 25a8z2 - 18a8z4 + 2a8z6 + 2a8z8 - a9z-1 + a9z + 3a9z3 - 5a9z5 + 2a9z7 + a10z-2 - 4a10 + 6a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     4   j = 1                   6 1   j = -1                 9 4     j = -3               10 7       j = -5             11 8         j = -7           8 10           j = -9         8 11             j = -11       5 11               j = -13     1 5                 j = -15   1 5                   j = -17   1                     j = -19 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, 390]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 390]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[18, 12, 19, 11], X[20, 13, 21, 14], X[22, 20, 9, 19], X[12, 21, 13, 22], > X[14, 18, 15, 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, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 6 10 16 19 21 18 15 2 -10 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 2 5 -20 -18 4 2 4 2 2 -1 + q + --- + --- + --- + --- - q - q + --- - --- + --- + --- + -- + 28 26 24 22 16 14 12 10 8 q q q q q q q q q 5 3 5 2 4 6 > -- - -- + -- - 2 q + 3 q - q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 390]][a, z]
Out[8]=	2 4 6 8 10 2 4 6 8 a 3 a 4 a 3 a a 2 2 4 2 3 a - 7 a + 8 a - 4 a + -- - ---- + ---- - ---- + --- + 4 a z - 7 a z + 2 2 2 2 2 z z z z z 6 2 4 2 4 4 4 2 6 > 6 a z - z + 2 a z - 4 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 390]][a, z]
Out[9]=	2 4 6 8 10 3 2 4 6 8 10 a 3 a 4 a 3 a a a -4 a - 14 a - 21 a - 14 a - 4 a + -- + ---- + ---- + ---- + --- - -- - 2 2 2 2 2 z z z z z z 5 7 9 a a a 3 5 7 9 2 2 4 2 6 2 > -- - -- - -- + a z + a z + a z + a z + 7 a z + 23 a z + 35 a z + z z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 25 a z + 6 a z + 3 a z + 10 a z + 10 a z + 6 a z + 3 a z - 5 4 2 4 4 4 6 4 8 4 10 4 z 5 > 5 z - 3 a z - 3 a z - 19 a z - 18 a z - 4 a z + -- - 15 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 22 a z - 12 a z - 11 a z - 5 a z + 5 z - 12 a z - 23 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 5 a z + 2 a z + a z + 10 a z + 6 a z - 3 a z + 3 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 2 a z + 10 a z + 14 a z + 6 a z + 2 a z + 5 a z + 7 a z + 7 9 4 10 6 10 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 1 1 5 1 5 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 8 11 8 10 11 8 10 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a390

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