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

K11n4 K11n6

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X20,16,21,15 X17,12,18,13 X22,20,1,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 7t-2 + 17t-1 - 21 + 17t - 7t2 + t3

Conway Polynomial:

1 - 2z2 - z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{1041, ...}

Determinant and Signature:

{71, 2}

Jones Polynomial:

2q-1 - 4 + 8q - 11q2 + 12q3 - 12q4 + 10q5 - 7q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 + q-2 + 3q2 - 3q4 - q8 - 2q10 + 2q12 - 2q14 + 3q16 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + 2a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - 4a-2 - 7a-2z2 - 3a-2z4 + 3 + 2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 7a-8z4 + 4a-8z6 - a-7z + 4a-7z3 - 11a-7z5 + 6a-7z7 + 6a-6z2 - 10a-6z4 - a-6z6 + 4a-6z8 - 5a-5z + 17a-5z3 - 22a-5z5 + 9a-5z7 + a-5z9 + 2a-4 - 2a-4z2 + 4a-4z4 - 8a-4z6 + 6a-4z8 - 5a-3z + 11a-3z3 - 9a-3z5 + 4a-3z7 + a-3z9 + 4a-2 - 12a-2z2 + 10a-2z4 - 3a-2z6 + 2a-2z8 - a-1z - a-1z3 + a-1z5 + a-1z7 + 3 - 6z2 + 3z4

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

{-2, -2}

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=2 is the signature of 115. 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 = 17                   1 j = 15                 3   j = 13               4 1   j = 11             6 3     j = 9           6 4       j = 7         6 6         j = 5       5 6           j = 3     3 6             j = 1   2 6               j = -1   2                 j = -3 2

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, 5]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 5]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[11, 18, 12, 19], X[6, 14, 7, 13], X[20, 16, 21, 15], > X[17, 12, 18, 13], X[22, 20, 1, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 5]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, 8, -11, -9, 6, 10, > -8, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 5]]
Out[5]=	DTCode[4, 8, 10, 14, 2, -18, 6, 20, -12, 22, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 5]][t]
Out[6]=	-3 7 17 2 3 -21 + t - -- + -- + 17 t - 7 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 5]][z]
Out[7]=	2 4 6 1 - 2 z - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 41], Knot[11, NonAlternating, 5]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 5]], KnotSignature[Knot[11, NonAlternating, 5]]}
Out[9]=	{71, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 5]][q]
Out[10]=	2 2 3 4 5 6 7 8 -4 + - + 8 q - 11 q + 12 q - 12 q + 10 q - 7 q + 4 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 5]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 5]][q]
Out[12]=	2 -2 2 4 8 10 12 14 16 20 22 24 -- + q + 3 q - 3 q - q - 2 q + 2 q - 2 q + 3 q - q + 2 q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 5]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 4 2 z 4 z 7 z z 3 z 3 z z 3 + -- - -- + 2 z - -- + ---- - ---- - -- + ---- - ---- + -- 4 2 6 4 2 6 4 2 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 5]][a, z]
Out[14]=	2 2 2 2 3 2 4 z 5 z 5 z z 2 2 z 6 z 2 z 12 z z 3 + -- + -- - -- - --- - --- - - - 6 z + ---- + ---- - ---- - ----- - -- + 4 2 7 5 3 a 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 4 5 4 z 17 z 11 z z 4 7 z 10 z 4 z 10 z z > ---- + ----- + ----- - -- + 3 z - ---- - ----- + ---- + ----- + -- - 7 5 3 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 11 z 22 z 9 z z 4 z z 8 z 3 z 6 z 9 z 4 z > ----- - ----- - ---- + -- + ---- - -- - ---- - ---- + ---- + ---- + ---- + 7 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a a 7 8 8 8 9 9 z 4 z 6 z 2 z z z > -- + ---- + ---- + ---- + -- + -- a 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 5]], Vassiliev[3][Knot[11, NonAlternating, 5]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 5]][q, t]
Out[16]=	3 2 2 2 q 3 5 5 2 7 2 6 q + 3 q + ----- + --- + --- + 6 q t + 5 q t + 6 q t + 6 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 6 q t + 6 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 15 6 17 7 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n5

K11n4 K11n6