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

K11a145 K11a147

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X16,5,17,6 X18,7,19,8 X14,10,15,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X6,17,7,18 X12,20,13,19 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 14 2 20 22 6 12 8

Alexander Polynomial:

- t-4 + 6t-3 - 15t-2 + 25t-1 - 29 + 25t - 15t2 + 6t3 - t4

Conway Polynomial:

1 + 3z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{123, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 17q-4 + 20q-3 - 19q-2 + 18q-1 - 13 + 8q - 4q2 + q3

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

{K11a294, ...}

A2 (sl(3)) Invariant:

- q-24 - 2q-18 + 3q-16 - 3q-14 + q-12 + 2q-10 - q-8 + 6q-6 - 3q-4 + 3q-2 - 1 - 2q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2z2 + 3z4 + z6 - 6a2z2 - 9a2z4 - 5a2z6 - a2z8 + 3a4 + 10a4z2 + 8a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 5a-1z3 - 10a-1z5 + 4a-1z7 - 5z2 + 15z4 - 19z6 + 7z8 + 3az3 + az5 - 11az7 + 6az9 - 14a2z2 + 37a2z4 - 35a2z6 + 9a2z8 + 2a2z10 - 3a3z3 + 18a3z5 - 24a3z7 + 11a3z9 + 3a4 - 16a4z2 + 31a4z4 - 26a4z6 + 8a4z8 + 2a4z10 - 2a5z + 4a5z3 - a5z5 - 4a5z7 + 5a5z9 + 2a6 - 5a6z2 + 6a6z4 - 8a6z6 + 6a6z8 - a7z + 3a7z3 - 7a7z5 + 5a7z7 + 2a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{3, -5}

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 11146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   5 1   j = 1                 8 3     j = -1               10 5       j = -3             10 9         j = -5           10 9           j = -7         7 10             j = -9       5 10               j = -11     2 7                 j = -13   1 5                   j = -15   2                     j = -17 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, Alternating, 146]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 146]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[18, 7, 19, 8], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[6, 17, 7, 18], X[12, 20, 13, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 146]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -11, 5, -2, 6, -10, 7, -5, 8, -3, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 146]]
Out[5]=	DTCode[4, 10, 16, 18, 14, 2, 20, 22, 6, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 146]][t]
Out[6]=	-4 6 15 25 2 3 4 -29 - t + -- - -- + -- + 25 t - 15 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 146]][z]
Out[7]=	2 4 6 8 1 + 3 z + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 146]}
In[9]:=	{KnotDet[Knot[11, Alternating, 146]], KnotSignature[Knot[11, Alternating, 146]]}
Out[9]=	{123, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 146]][q]
Out[10]=	-8 3 7 12 17 20 19 18 2 3 -13 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 146], Knot[11, Alternating, 294]}
In[12]:=	A2Invariant[Knot[11, Alternating, 146]][q]
Out[12]=	-24 2 3 3 -12 2 -8 6 3 3 2 4 -1 - q - --- + --- - --- + q + --- - q + -- - -- + -- - 2 q + 2 q - 18 16 14 10 6 4 2 q q q q q q q 6 8 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 146]][a, z]
Out[13]=	4 6 2 2 2 4 2 6 2 4 2 4 4 4 3 a - 2 a + 2 z - 6 a z + 10 a z - 3 a z + 3 z - 9 a z + 8 a z - 6 4 6 2 6 4 6 2 8 > a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 146]][a, z]
Out[14]=	4 6 5 7 9 2 2 2 4 2 6 2 3 a + 2 a - 2 a z - a z + a z - 5 z - 14 a z - 16 a z - 5 a z + 3 8 2 5 z 3 3 3 5 3 7 3 9 3 4 > 2 a z + ---- + 3 a z - 3 a z + 4 a z + 3 a z - 2 a z + 15 z - a 4 5 2 z 2 4 4 4 6 4 8 4 10 z 5 3 5 > ---- + 37 a z + 31 a z + 6 a z - 5 a z - ----- + a z + 18 a z - 2 a a 6 5 5 7 5 9 5 6 z 2 6 4 6 6 6 > a z - 7 a z + a z - 19 z + -- - 35 a z - 26 a z - 8 a z + 2 a 7 8 6 4 z 7 3 7 5 7 7 7 8 2 8 > 3 a z + ---- - 11 a z - 24 a z - 4 a z + 5 a z + 7 z + 9 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 8 a z + 6 a z + 6 a z + 11 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 146]], Vassiliev[3][Knot[11, Alternating, 146]]}
Out[15]=	{3, -5}
In[16]:=	Kh[Knot[11, Alternating, 146]][q, t]
Out[16]=	9 10 1 2 1 5 2 7 5 10 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 10 10 9 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a146

K11a145 K11a147