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

K11a156 K11a158

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 12 16 2 20 8 22 6 14

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 28t-1 - 33 + 28t - 16t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a264, K11a305, ...}

Determinant and Signature:

{135, -2}

Jones Polynomial:

- q-8 + 3q-7 - 8q-6 + 14q-5 - 18q-4 + 22q-3 - 22q-2 + 19q-1 - 14 + 9q - 4q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-20 - 3q-18 + 4q-16 - q-14 + 4q-12 + 3q-10 - 3q-8 + 3q-6 - 6q-4 + 3q-2 - q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2 + 3z2 + 3z4 + z6 - 5a2 - 10a2z2 - 10a2z4 - 5a2z6 - a2z8 + 7a4 + 12a4z2 + 8a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 - a-1z + 5a-1z3 - 9a-1z5 + 4a-1z7 + 2 - 4z2 + 10z4 - 16z6 + 7z8 - 2az + 8az3 - 7az5 - 7az7 + 6az9 + 5a2 - 20a2z2 + 38a2z4 - 39a2z6 + 12a2z8 + 2a2z10 - 3a3z + 8a3z3 + 4a3z5 - 20a3z7 + 12a3z9 + 7a4 - 28a4z2 + 46a4z4 - 39a4z6 + 13a4z8 + 2a4z10 - 5a5z + 13a5z3 - 8a5z5 - 3a5z7 + 6a5z9 + 3a6 - 12a6z2 + 16a6z4 - 14a6z6 + 8a6z8 - 2a7z + 6a7z3 - 9a7z5 + 6a7z7 + a8z2 - 4a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{2, -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 11157. 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                   6 1   j = 1                 8 3     j = -1               11 6       j = -3             12 9         j = -5           10 10           j = -7         8 12             j = -9       6 10               j = -11     2 8                 j = -13   1 6                   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, 157]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 157]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[12, 8, 13, 7], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 157]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -4, 7, -11, 8, -5, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 157]]
Out[5]=	DTCode[4, 10, 18, 12, 16, 2, 20, 8, 22, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 157]][t]
Out[6]=	-4 6 16 28 2 3 4 -33 - t + -- - -- + -- + 28 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 157]][z]
Out[7]=	2 6 8 1 + 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 157], Knot[11, Alternating, 264], > Knot[11, Alternating, 305]}
In[9]:=	{KnotDet[Knot[11, Alternating, 157]], KnotSignature[Knot[11, Alternating, 157]]}
Out[9]=	{135, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 157]][q]
Out[10]=	-8 3 8 14 18 22 22 19 2 3 -14 - q + -- - -- + -- - -- + -- - -- + -- + 9 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, 157]}
In[12]:=	A2Invariant[Knot[11, Alternating, 157]][q]
Out[12]=	-24 -20 3 4 -14 4 3 3 3 6 3 2 4 -q - q - --- + --- - q + --- + --- - -- + -- - -- + -- - q + 3 q - 18 16 12 10 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 157]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 - 5 a + 7 a - 3 a + 3 z - 10 a z + 12 a z - 3 a z + 3 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 10 a z + 8 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 157]][a, z]
Out[14]=	2 4 6 z 3 5 7 9 2 2 + 5 a + 7 a + 3 a - - - 2 a z - 3 a z - 5 a z - 2 a z + a z - 4 z + a 2 3 z 2 2 4 2 6 2 8 2 5 z 3 3 3 > -- - 20 a z - 28 a z - 12 a z + a z + ---- + 8 a z + 8 a z + 2 a a 4 5 3 7 3 9 3 4 2 z 2 4 4 4 > 13 a z + 6 a z - 2 a z + 10 z - ---- + 38 a z + 46 a z + 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 9 5 > 16 a z - 4 a z - ---- - 7 a z + 4 a z - 8 a z - 9 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 4 z 7 > 16 z + -- - 39 a z - 39 a z - 14 a z + 3 a z + ---- - 7 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 20 a z - 3 a z + 6 a z + 7 z + 12 a z + 13 a z + 8 a z + 9 3 9 5 9 2 10 4 10 > 6 a z + 12 a z + 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 157]], Vassiliev[3][Knot[11, Alternating, 157]]}
Out[15]=	{2, -5}
In[16]:=	Kh[Knot[11, Alternating, 157]][q, t]
Out[16]=	9 11 1 2 1 6 2 8 6 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 8 12 10 10 12 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 6 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 K11a157

K11a156 K11a158