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

K11a155 K11a157

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 12 14 2 8 20 22 6 16

Alexander Polynomial:

- t-4 + 5t-3 - 11t-2 + 18t-1 - 21 + 18t - 11t2 + 5t3 - t4

Conway Polynomial:

1 + 3z2 - z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, -2}

Jones Polynomial:

- q-8 + 2q-7 - 5q-6 + 9q-5 - 12q-4 + 15q-3 - 14q-2 + 13q-1 - 10 + 6q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-20 - 2q-18 + 3q-16 + 3q-12 + 3q-10 - q-8 + 3q-6 - 4q-4 + q-2 - 1 - q2 + 2q4 - q6 + q8

HOMFLY-PT Polynomial:

2 + 5z2 + 4z4 + z6 - 6a2 - 15a2z2 - 14a2z4 - 6a2z6 - a2z8 + 9a4 + 17a4z2 + 10a4z4 + 2a4z6 - 4a6 - 4a6z2 - a6z4

Kauffman Polynomial:

2a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 7a-1z3 - 9a-1z5 + 3a-1z7 + 2 - 4z2 + 6z4 - 10z6 + 4z8 - 5az + 10az3 - 8az5 - 3az7 + 3az9 + 6a2 - 22a2z2 + 32a2z4 - 24a2z6 + 6a2z8 + a2z10 - 6a3z + 8a3z3 + 7a3z5 - 13a3z7 + 6a3z9 + 9a4 - 30a4z2 + 43a4z4 - 25a4z6 + 6a4z8 + a4z10 - 6a5z + 10a5z3 - 4a5z7 + 3a5z9 + 4a6 - 13a6z2 + 16a6z4 - 10a6z6 + 4a6z8 - a7z + 2a7z3 - 5a7z5 + 3a7z7 + a8z2 - 4a8z4 + 2a8z6 + 2a9z - 3a9z3 + a9z5

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

{3, -7}

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 11156. 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                     2   j = 3                   4 1   j = 1                 6 2     j = -1               7 4       j = -3             8 7         j = -5           7 6           j = -7         5 8             j = -9       4 7               j = -11     1 5                 j = -13   1 4                   j = -15   1                     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, 156]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 156]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[12, 8, 13, 7], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 156]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -4, 7, -5, 8, -11, 9, -3, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 156]]
Out[5]=	DTCode[4, 10, 18, 12, 14, 2, 8, 20, 22, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 156]][t]
Out[6]=	-4 5 11 18 2 3 4 -21 - t + -- - -- + -- + 18 t - 11 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 156]][z]
Out[7]=	2 4 6 8 1 + 3 z - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 156]}
In[9]:=	{KnotDet[Knot[11, Alternating, 156]], KnotSignature[Knot[11, Alternating, 156]]}
Out[9]=	{91, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 156]][q]
Out[10]=	-8 2 5 9 12 15 14 13 2 3 -10 - q + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 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, 156]}
In[12]:=	A2Invariant[Knot[11, Alternating, 156]][q]
Out[12]=	-24 -22 -20 2 3 3 3 -8 3 4 -2 2 -1 - q - q - q - --- + --- + --- + --- - q + -- - -- + q - q + 18 16 12 10 6 4 q q q q q q 4 6 8 > 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 156]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 - 6 a + 9 a - 4 a + 5 z - 15 a z + 17 a z - 4 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 14 a z + 10 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 156]][a, z]
Out[14]=	2 4 6 2 z 3 5 7 9 2 2 + 6 a + 9 a + 4 a - --- - 5 a z - 6 a z - 6 a z - a z + 2 a z - 4 z + a 2 3 2 z 2 2 4 2 6 2 8 2 7 z 3 3 3 > ---- - 22 a z - 30 a z - 13 a z + a z + ---- + 10 a z + 8 a z + 2 a a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 > 10 a z + 2 a z - 3 a z + 6 z - ---- + 32 a z + 43 a z + 2 a 5 6 4 8 4 9 z 5 3 5 7 5 9 5 6 > 16 a z - 4 a z - ---- - 8 a z + 7 a z - 5 a z + a z - 10 z + a 6 7 z 2 6 4 6 6 6 8 6 3 z 7 3 7 > -- - 24 a z - 25 a z - 10 a z + 2 a z + ---- - 3 a z - 13 a z - 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 3 9 > 4 a z + 3 a z + 4 z + 6 a z + 6 a z + 4 a z + 3 a z + 6 a z + 5 9 2 10 4 10 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 156]], Vassiliev[3][Knot[11, Alternating, 156]]}
Out[15]=	{3, -7}
In[16]:=	Kh[Knot[11, Alternating, 156]][q, t]
Out[16]=	7 7 1 1 1 4 1 5 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 8 7 6 8 4 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 6 q t + 2 q t + 4 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 + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a156

K11a155 K11a157