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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-10 + 3q-9 - 6q-8 + 10q-7 - 13q-6 + 16q-5 - 15q-4 + 15q-3 - 10q-2 + 7q-1 - 3 + q

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 - q-26 + 4q-22 - 2q-20 + 2q-18 + 3q-16 + 2q-14 + 6q-12 + 2q-10 + 6q-8 + 3q-6 - q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2z-2 + 3a2 + 2a2z2 - a2z4 - 2a4z-2 - 4a4 - 4a4z2 - 3a4z4 + a6z-2 - a6z2 - 2a6z4 + 2a8 + 3a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 + a2z-2 - 4a2 + 8a2z2 - 8a2z4 + 6a2z6 - 2a3z-1 + 4a3z + a3z3 - 7a3z5 + 7a3z7 + 2a4z-2 - 7a4 + 16a4z2 - 13a4z4 - 2a4z6 + 6a4z8 - 2a5z-1 + 6a5z - 2a5z3 - 8a5z5 + 4a5z9 + a6z-2 - 2a6 - 7a6z2 + 23a6z4 - 30a6z6 + 10a6z8 + a6z10 + 2a7z3 + 3a7z5 - 15a7z7 + 7a7z9 + 4a8 - 21a8z2 + 42a8z4 - 34a8z6 + 7a8z8 + a8z10 - 4a9z + 12a9z3 - 3a9z5 - 7a9z7 + 3a9z9 + 2a10 - 7a10z2 + 15a10z4 - 12a10z6 + 3a10z8 - 2a11z + 5a11z3 - 4a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     2   j = -1                   5 1   j = -3                 6 3     j = -5               9 4       j = -7             8 8         j = -9           8 7           j = -11         6 9             j = -13       4 7               j = -15     2 6                 j = -17   1 4                   j = -19   2                     j = -21 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, 406]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 406]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[22, 15, 11, 16], > X[14, 7, 15, 8], X[20, 17, 21, 18], X[8, 20, 9, 19], X[18, 10, 19, 9], > X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -7, 8, -3}, > {11, -2, 3, -5, 4, -9, 6, -8, 7, -6, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-10 3 6 10 13 16 15 15 10 7 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 4 2 2 3 2 6 2 6 -1 - q - q + --- - q + --- - --- + --- + --- + --- + --- + --- + -- + 28 22 20 18 16 14 12 10 8 q q q q q q q q q 3 -4 4 2 4 > -- - q + -- - q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 406]][a, z]
Out[8]=	2 4 6 2 4 8 10 a 2 a a 2 2 2 4 2 6 2 3 a - 4 a + 2 a - a + -- - ---- + -- + z + 2 a z - 4 a z - a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 > 3 a z - a z - 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 406]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 -4 a - 7 a - 2 a + 4 a + 2 a + -- + ---- + -- - ---- - ---- + 4 a z + 2 2 2 z z z z z 5 9 11 2 2 2 4 2 6 2 8 2 > 6 a z - 4 a z - 2 a z - z + 8 a z + 16 a z - 7 a z - 21 a z - 10 2 3 3 3 5 3 7 3 9 3 11 3 4 > 7 a z - 2 a z + a z - 2 a z + 2 a z + 12 a z + 5 a z + z - 2 4 4 4 6 4 8 4 10 4 5 3 5 > 8 a z - 13 a z + 23 a z + 42 a z + 15 a z + 3 a z - 7 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 8 a z + 3 a z - 3 a z - 4 a z + 6 a z - 2 a z - 30 a z - 8 6 10 6 3 7 7 7 9 7 11 7 4 8 > 34 a z - 12 a z + 7 a z - 15 a z - 7 a z + a z + 6 a z + 6 8 8 8 10 8 5 9 7 9 9 9 6 10 > 10 a z + 7 a z + 3 a z + 4 a z + 7 a z + 3 a z + a z + 8 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 4 2 6 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 7 6 9 8 7 8 8 9 4 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 6 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a406

L11a405 L11a407