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K11n89 K11n91

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X13,21,14,20 X8,16,9,15 X6,18,7,17 X19,1,20,22 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 -20 8 6 -22 -14

Alexander Polynomial:

- 2t-3 + 7t-2 - 8t-1 + 7 - 8t + 7t2 - 2t3

Conway Polynomial:

1 + 2z2 - 5z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{41, 4}

Jones Polynomial:

1 - 2q + 4q2 - 5q3 + 7q4 - 7q5 + 6q6 - 5q7 + 3q8 - q9

Other knots (up to mirrors) with the same Jones Polynomial:

{920, ...}

A2 (sl(3)) Invariant:

1 + q6 + 3q10 + q14 - q16 - 3q18 - q22 + 2q24 + q26 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 4a-8z2 + a-8z4 - 4a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 2a-4 - 3a-4z4 - a-4z6 + a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- a-11z + a-11z3 + a-10 - 3a-10z2 + 3a-10z4 - a-9z + 3a-9z3 - a-9z5 + a-9z7 + 3a-8 - 13a-8z2 + 18a-8z4 - 8a-8z6 + 2a-8z8 - a-7z - a-7z3 + 3a-7z5 - 2a-7z7 + a-7z9 + 4a-6 - 17a-6z2 + 23a-6z4 - 15a-6z6 + 4a-6z8 - a-5z + 2a-5z3 - 3a-5z5 - a-5z7 + a-5z9 + 2a-4 - 3a-4z2 + 4a-4z4 - 6a-4z6 + 2a-4z8 + 5a-3z3 - 7a-3z5 + 2a-3z7 - a-2 + 4a-2z2 - 4a-2z4 + a-2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, 4}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=4 is the signature of 1190. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                   1 j = 17                 2   j = 15               3 1   j = 13             3 2     j = 11           4 3       j = 9         3 3         j = 7       2 4           j = 5     2 3             j = 3   1 3               j = 1   1                 j = -1 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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 90]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 90]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 11, 3, 12], X[13, 21, 14, 20], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[19, 1, 20, 22], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 90]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, -7, 11, 8, -4, 9, -5, -10, > 7, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 90]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, -20, 8, 6, -22, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 90]][t]
Out[6]=	2 7 8 2 3 7 - -- + -- - - - 8 t + 7 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 90]][z]
Out[7]=	2 4 6 1 + 2 z - 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 90]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 90]], KnotSignature[Knot[11, NonAlternating, 90]]}
Out[9]=	{41, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 90]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 4 q - 5 q + 7 q - 7 q + 6 q - 5 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 20], Knot[11, NonAlternating, 90]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 90]][q]
Out[12]=	6 10 14 16 18 22 24 26 32 1 + q + 3 q + q - q - 3 q - q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 90]][a, z]
Out[13]=	2 2 2 4 4 4 4 6 -10 3 4 2 -2 4 z 5 z 3 z z 4 z 3 z z z -a + -- - -- + -- + a + ---- - ---- + ---- + -- - ---- - ---- + -- - -- - 8 6 4 8 6 2 8 6 4 2 6 a a a a a a a a a a a 6 z > -- 4 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 90]][a, z]
Out[14]=	2 2 2 2 -10 3 4 2 -2 z z z z 3 z 13 z 17 z 3 z a + -- + -- + -- - a - --- - -- - -- - -- - ---- - ----- - ----- - ---- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 4 4 z z 3 z z 2 z 5 z 3 z 18 z 23 z 4 z 4 z > ---- + --- + ---- - -- + ---- + ---- + ---- + ----- + ----- + ---- - ---- - 2 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 z 3 z 3 z 7 z 8 z 15 z 6 z z z 2 z z > -- + ---- - ---- - ---- - ---- - ----- - ---- + -- + -- - ---- - -- + 9 7 5 3 8 6 4 2 9 7 5 a a a a a a a a a a a 7 8 8 8 9 9 2 z 2 z 4 z 2 z z z > ---- + ---- + ---- + ---- + -- + -- 3 8 6 4 7 5 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 90]], Vassiliev[3][Knot[11, NonAlternating, 90]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 90]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 3 q + 2 q + ---- + - + -- + 3 q t + 2 q t + 4 q t + 3 q t + 3 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 17 6 > 4 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 19 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n90

K11n89 K11n91