© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n44 K11n46

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X9,21,10,20 X11,18,12,19 X6,13,7,14 X15,10,16,11 X17,1,18,22 X19,14,20,15 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-2 - 6t-1 + 9 - 6t + 2t2

Conway Polynomial:

1 + 2z2 + 2z4

Other knots with the same Alexander/Conway Polynomial:

{88, 10129, K11n39, K11n50, K11n132, ...}

Determinant and Signature:

{25, 0}

Jones Polynomial:

- q-4 + q-3 - q-1 + 4 - 4q + 5q2 - 5q3 + 4q4 - 3q5 + q6

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

{K11n39, ...}

A2 (sl(3)) Invariant:

- q-12 - q-10 - q-8 + q-6 + 3q-2 + 3 + q2 + 2q4 - 2q6 - 2q10 - q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 5a-2 - 9a-2z2 - 5a-2z4 - a-2z6 + 8 + 13z2 + 7z4 + z6 - 3a2 - 4a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 8a-5z3 - 11a-5z5 + 3a-5z7 + a-4 - 3a-4z2 + 8a-4z4 - 11a-4z6 + 3a-4z8 - 8a-3z + 20a-3z3 - 14a-3z5 + a-3z9 + 5a-2 - 18a-2z2 + 30a-2z4 - 20a-2z6 + 4a-2z8 - 12a-1z + 23a-1z3 - 7a-1z5 - 3a-1z7 + a-1z9 + 8 - 24z2 + 32z4 - 15z6 + 2z8 - 10az + 20az3 - 10az5 + az7 + 3a2 - 10a2z2 + 13a2z4 - 7a2z6 + a2z8 - 4a3z + 9a3z3 - 6a3z5 + a3z7

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

{2, -1}

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=0 is the signature of 1145. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     2   j = 9                   2 1   j = 7                 3 2     j = 5             1 3 2       j = 3             2 3         j = 1         1 4 3           j = -1       1 1 3             j = -3       1 2               j = -5   1 1 1                 j = -7                         j = -9 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, 45]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 45]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[9, 21, 10, 20], X[11, 18, 12, 19], X[6, 13, 7, 14], X[15, 10, 16, 11], > X[17, 1, 18, 22], X[19, 14, 20, 15], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 45]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 6, -10, > 5, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 45]]
Out[5]=	DTCode[4, 8, 12, 2, -20, -18, 6, -10, -22, -14, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 45]][t]
Out[6]=	2 6 2 9 + -- - - - 6 t + 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 45]][z]
Out[7]=	2 4 1 + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 8], Knot[10, 129], Knot[11, NonAlternating, 39], > Knot[11, NonAlternating, 45], Knot[11, NonAlternating, 50], > Knot[11, NonAlternating, 132]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 45]], KnotSignature[Knot[11, NonAlternating, 45]]}
Out[9]=	{25, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 45]][q]
Out[10]=	-4 -3 1 2 3 4 5 6 4 - q + q - - - 4 q + 5 q - 5 q + 4 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 39], Knot[11, NonAlternating, 45]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 45]][q]
Out[12]=	-12 -10 -8 -6 3 2 4 6 10 12 14 16 3 - q - q - q + q + -- + q + 2 q - 2 q - 2 q - q + q - q + 2 q 18 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 45]][a, z]
Out[13]=	2 2 4 4 -4 5 2 2 2 z 9 z 2 2 4 z 5 z 8 + a - -- - 3 a + 13 z + ---- - ---- - 4 a z + 7 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 z > a z + z - -- 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 45]][a, z]
Out[14]=	2 2 -4 5 2 2 z 8 z 12 z 3 2 z 3 z 8 + a + -- + 3 a - --- - --- - ---- - 10 a z - 4 a z - 24 z + -- - ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 18 z 2 2 8 z 20 z 23 z 3 3 3 4 > ----- - 10 a z + ---- + ----- + ----- + 20 a z + 9 a z + 32 z - 2 5 3 a a a a 4 4 4 5 5 5 3 z 8 z 30 z 2 4 11 z 14 z 7 z 5 3 5 > ---- + ---- + ----- + 13 a z - ----- - ----- - ---- - 10 a z - 6 a z - 6 4 2 5 3 a a a a a a 6 6 6 7 7 6 z 11 z 20 z 2 6 3 z 3 z 7 3 7 8 > 15 z + -- - ----- - ----- - 7 a z + ---- - ---- + a z + a z + 2 z + 6 4 2 5 a a a a a 8 8 9 9 3 z 4 z 2 8 z z > ---- + ---- + a z + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 45]], Vassiliev[3][Knot[11, NonAlternating, 45]]}
Out[15]=	{2, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 45]][q, t]
Out[16]=	3 1 1 1 1 1 1 2 1 q - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + - + q 9 5 5 4 5 3 5 2 3 2 2 3 q t t q t q t q t q t q t q t q t 3 5 3 2 5 2 5 3 7 3 7 4 > 3 q t + 2 q t + q t + 3 q t + 3 q t + 2 q t + 3 q t + 2 q t + 9 4 9 5 11 5 13 6 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n45

K11n44 K11n46