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

K11n144 K11n146

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X5,11,6,10 X7,20,8,21 X9,1,10,22 X18,11,19,12 X2,14,3,13 X15,9,16,8 X17,6,18,7 X12,19,13,20 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + 2z2 + 5z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

- q-4 + q-3 + 2 - 2q + 2q2 - 2q3 + 2q4 - q5

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

{...}

A2 (sl(3)) Invariant:

- q-12 - q-10 + q-6 + 2q-2 + 2 + q2 + q4 - q6 - q10 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 - 2a-2 - 2a-2z2 + 5 + 9z2 + 6z4 + z6 - 2a2 - 4a2z2 - a2z4

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 + 4a-4z2 - 7a-4z4 + 2a-4z6 + a-3z3 - 3a-3z5 + a-3z7 + 2a-2 - 2a-2z2 - 2a-2z4 + a-2z6 - 4a-1z + 10a-1z3 - 6a-1z5 + a-1z7 + 5 - 16z2 + 19z4 - 8z6 + z8 - 6az + 15az3 - 8az5 + az7 + 2a2 - 10a2z2 + 14a2z4 - 7a2z6 + a2z8 - 3a3z + 9a3z3 - 6a3z5 + a3z7

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

{2, 0}

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 11145. 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 j = 11                     1 j = 9                   1   j = 7                 1 1   j = 5             1 2 1     j = 3             1 1       j = 1         1 3 2         j = -1       1 1 2           j = -3       1 1             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, 145]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 145]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[5, 11, 6, 10], X[7, 20, 8, 21], > X[9, 1, 10, 22], X[18, 11, 19, 12], X[2, 14, 3, 13], X[15, 9, 16, 8], > X[17, 6, 18, 7], X[12, 19, 13, 20], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 145]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 9, -4, 8, -5, 3, 6, -10, 7, -2, -8, 11, -9, -6, 10, > 4, -11, 5]
In[5]:=	DTCode[Knot[11, NonAlternating, 145]]
Out[5]=	DTCode[4, 14, -10, -20, -22, 18, 2, -8, -6, 12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 145]][t]
Out[6]=	-3 -2 3 2 3 7 + t - t - - - 3 t - t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 145]][z]
Out[7]=	2 4 6 1 + 2 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 145]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 145]], KnotSignature[Knot[11, NonAlternating, 145]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 145]][q]
Out[10]=	-4 -3 2 3 4 5 2 - q + q - 2 q + 2 q - 2 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 145]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 145]][q]
Out[12]=	-12 -10 -6 2 2 4 6 10 14 16 2 - q - q + q + -- + q + q - q - q + q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 145]][a, z]
Out[13]=	2 2 2 2 2 z 2 z 2 2 4 2 4 6 5 - -- - 2 a + 9 z - -- - ---- - 4 a z + 6 z - a z + z 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 145]][a, z]
Out[14]=	2 2 2 2 z 4 z 3 2 4 z 2 z 2 2 5 + -- + 2 a + -- - --- - 6 a z - 3 a z - 16 z + ---- - ---- - 10 a z - 2 5 a 4 2 a a a a 3 3 3 4 4 3 z z 10 z 3 3 3 4 7 z 2 z 2 4 > ---- + -- + ----- + 15 a z + 9 a z + 19 z - ---- - ---- + 14 a z + 5 3 a 4 2 a a a a 5 5 5 6 6 7 z 3 z 6 z 5 3 5 6 2 z z 2 6 z > -- - ---- - ---- - 8 a z - 6 a z - 8 z + ---- + -- - 7 a z + -- + 5 3 a 4 2 3 a a a a a 7 z 7 3 7 8 2 8 > -- + a z + a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 145]], Vassiliev[3][Knot[11, NonAlternating, 145]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 145]][q, t]
Out[16]=	2 1 1 1 1 1 1 1 1 q - + 3 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 9 4 > 2 q t + q t + q t + q t + 2 q t + q t + q t + q t + q t + 11 5 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n145

K11n144 K11n146