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

K11n12 K11n14

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X11,19,12,18 X13,21,14,20 X6,16,7,15 X17,1,18,22 X19,13,20,12 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 3t-3 - 2t-2 + t-1 - 1 + t - 2t2 + 3t3 - t4

Conway Polynomial:

1 + 4z2 - 4z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{15, 6}

Jones Polynomial:

q - q2 + 2q3 - 2q4 + 2q5 - 2q6 + 2q7 - q8 + q9 - q10

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

{92, ...}

A2 (sl(3)) Invariant:

q4 + q6 + q8 + q10 - q16 - q20 + q22 + q24 + q26 - q30 - q34

HOMFLY-PT Polynomial:

- 2a-10 - a-10z2 + 5a-8 + 10a-8z2 + 6a-8z4 + a-8z6 - 6a-6 - 15a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + 4a-4 + 10a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-13z + a-12z2 - a-11z + a-11z3 + 2a-10 - 6a-10z2 + 2a-10z4 - a-9z + 3a-9z3 - 4a-9z5 + a-9z7 + 5a-8 - 19a-8z2 + 23a-8z4 - 12a-8z6 + 2a-8z8 + a-7z - 2a-7z3 + 6a-7z5 - 5a-7z7 + a-7z9 + 6a-6 - 26a-6z2 + 37a-6z4 - 19a-6z6 + 3a-6z8 - 4a-5z3 + 10a-5z5 - 6a-5z7 + a-5z9 + 4a-4 - 14a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8

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

{4, 10}

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=6 is the signature of 1113. 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 = 21                   1 j = 19                     j = 17               1 1   j = 15             1       j = 13           1 1       j = 11         1 1         j = 9       1 1           j = 7     1 1             j = 5   1 2               j = 3                     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, 13]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 13]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 21, 14, 20], X[6, 16, 7, 15], > X[17, 1, 18, 22], X[19, 13, 20, 12], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 13]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, 11, 8, -4, -9, 6, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 13]]
Out[5]=	DTCode[4, 8, 10, 16, 2, -18, -20, 6, -22, -12, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 13]][t]
Out[6]=	-4 3 2 1 2 3 4 -1 - t + -- - -- + - + t - 2 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 13]][z]
Out[7]=	2 4 6 8 1 + 4 z - 4 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 13]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 13]], KnotSignature[Knot[11, NonAlternating, 13]]}
Out[9]=	{15, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 13]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 q - q + 2 q - 2 q + 2 q - 2 q + 2 q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 2], Knot[11, NonAlternating, 13]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 13]][q]
Out[12]=	4 6 8 10 16 20 22 24 26 30 34 q + q + q + q - q - q + q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 13]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -2 5 6 4 z 10 z 15 z 10 z 6 z 16 z 6 z z --- + -- - -- + -- - --- + ----- - ----- + ----- + ---- - ----- + ---- + -- - 10 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 13]][a, z]
Out[14]=	2 2 2 2 2 2 5 6 4 z z z z z 6 z 19 z 26 z 14 z --- + -- + -- + -- + --- - --- - -- + -- + --- - ---- - ----- - ----- - ----- + 10 8 6 4 13 11 9 7 12 10 8 6 4 a a a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 z 3 z 2 z 4 z 2 z 23 z 37 z 16 z 4 z 6 z > --- + ---- - ---- - ---- + ---- + ----- + ----- + ----- - ---- + ---- + 11 9 7 5 10 8 6 4 9 7 a a a a a a a a a a 5 6 6 6 7 7 7 8 8 8 9 9 10 z 12 z 19 z 7 z z 5 z 6 z 2 z 3 z z z z > ----- - ----- - ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 5 8 6 4 9 7 5 8 6 4 7 5 a a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 13]], Vassiliev[3][Knot[11, NonAlternating, 13]]}
Out[15]=	{4, 10}
In[16]:=	Kh[Knot[11, NonAlternating, 13]][q, t]
Out[16]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 13 4 2 q + q + -- + -- + q t + q t + q t + q t + q t + q t + q t + 2 t t 15 4 17 5 17 6 21 7 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n13

K11n12 K11n14