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

K11n155 K11n157

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X12,6,13,5 X14,7,15,8 X16,10,17,9 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X2,19,3,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 18t-1 + 25 - 18t + 7t2 - t3

Conway Polynomial:

1 + z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1071, K11n179, ...}

Determinant and Signature:

{77, 0}

Jones Polynomial:

- 2q-3 + 6q-2 - 9q-1 + 12 - 13q + 13q2 - 10q3 + 7q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- 2q-10 + q-8 + 2q-6 - 2q-4 + 3q-2 - 1 + q2 + 2q4 - q6 + 3q8 - 3q10 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - a-2 - 4a-2z2 - 3a-2z4 - a-2z6 + 3 + 6z2 + 3z4 - a2 - 2a2z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 6a-5z3 - 11a-5z5 + 4a-5z7 - 3a-4z2 + 14a-4z4 - 18a-4z6 + 6a-4z8 - a-3z + 11a-3z3 - 14a-3z5 - a-3z7 + 3a-3z9 + a-2 - 12a-2z2 + 33a-2z4 - 35a-2z6 + 12a-2z8 - 2a-1z + 5a-1z3 - 6a-1z5 - a-1z7 + 3a-1z9 + 3 - 15z2 + 23z4 - 15z6 + 6z8 - 2az + 3az3 - 3az5 + 4az7 + a2 - 6a2z2 + 6a2z4 + a2z6 - a3z + 3a3z3

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

{1, 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 11156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 3   j = 9               4 1   j = 7             6 3     j = 5           7 4       j = 3         6 6         j = 1       6 7           j = -1     4 7             j = -3   2 5               j = -5   4                 j = -7 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, 156]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 156]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[11, 19, 12, 18], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[17, 4, 18, 5], X[2, 19, 3, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 156]]
Out[4]=	GaussCode[1, -10, -2, 9, 3, -1, 4, -11, 5, 2, -6, -3, 7, -4, 8, -5, -9, 6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 156]]
Out[5]=	DTCode[6, -10, 12, 14, 16, -18, 22, 20, -4, 2, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 156]][t]
Out[6]=	-3 7 18 2 3 25 - t + -- - -- - 18 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 156]][z]
Out[7]=	2 4 6 1 + z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 71], Knot[11, NonAlternating, 156], Knot[11, NonAlternating, 179]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 156]], KnotSignature[Knot[11, NonAlternating, 156]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 156]][q]
Out[10]=	2 6 9 2 3 4 5 6 12 - -- + -- - - - 13 q + 13 q - 10 q + 7 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 156]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 156]][q]
Out[12]=	2 -8 2 2 3 2 4 6 8 10 14 16 -1 - --- + q + -- - -- + -- + q + 2 q - q + 3 q - 3 q + q - 2 q + 10 6 4 2 q q q q 18 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 156]][a, z]
Out[13]=	2 2 4 4 6 -2 2 2 z 4 z 2 2 4 z 3 z z 3 - a - a + 6 z + -- - ---- - 2 a z + 3 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 156]][a, z]
Out[14]=	2 2 -2 2 z 2 z 3 2 3 z 12 z 2 2 3 + a + a - -- - --- - 2 a z - a z - 15 z - ---- - ----- - 6 a z + 3 a 4 2 a a a 3 3 3 4 4 4 6 z 11 z 5 z 3 3 3 4 2 z 14 z 33 z > ---- + ----- + ---- + 3 a z + 3 a z + 23 z - ---- + ----- + ----- + 5 3 a 6 4 2 a a a a a 5 5 5 6 6 6 2 4 11 z 14 z 6 z 5 6 z 18 z 35 z > 6 a z - ----- - ----- - ---- - 3 a z - 15 z + -- - ----- - ----- + 5 3 a 6 4 2 a a a a a 7 7 7 8 8 9 9 2 6 4 z z z 7 8 6 z 12 z 3 z 3 z > a z + ---- - -- - -- + 4 a z + 6 z + ---- + ----- + ---- + ---- 5 3 a 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 156]], Vassiliev[3][Knot[11, NonAlternating, 156]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 156]][q, t]
Out[16]=	7 2 4 2 5 4 3 3 2 - + 6 q + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + 6 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n156

K11n155 K11n157