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

K11a157 K11a159

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 2 16 20 8 22 6 12

Alexander Polynomial:

- t-4 + 5t-3 - 14t-2 + 25t-1 - 29 + 25t - 14t2 + 5t3 - t4

Conway Polynomial:

1 - 2z2 - 4z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a34, ...}

Determinant and Signature:

{119, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 12q-4 + 16q-3 - 19q-2 + 19q-1 - 16 + 13q - 8q2 + 4q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 3q-16 - q-14 - q-12 + 2q-10 - 5q-8 + 2q-6 - 3q-4 + q-2 + 3 - q2 + 4q4 - q6 + q10 - q12

HOMFLY-PT Polynomial:

- a-2 - 2a-2z2 - a-2z4 + 6 + 11z2 + 8z4 + 2z6 - 7a2 - 17a2z2 - 15a2z4 - 6a2z6 - a2z8 + 3a4 + 6a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- a-3z + 3a-3z3 - 3a-3z5 + a-3z7 + a-2 - 6a-2z2 + 15a-2z4 - 14a-2z6 + 4a-2z8 - 3a-1z + 9a-1z3 + 2a-1z5 - 13a-1z7 + 5a-1z9 + 6 - 27z2 + 56z4 - 46z6 + 8z8 + 2z10 - 5az + 10az3 + 10az5 - 30az7 + 12az9 + 7a2 - 34a2z2 + 65a2z4 - 57a2z6 + 14a2z8 + 2a2z10 - 7a3z + 18a3z3 - 12a3z5 - 7a3z7 + 7a3z9 + 3a4 - 9a4z2 + 17a4z4 - 19a4z6 + 10a4z8 - 4a5z + 12a5z3 - 14a5z5 + 9a5z7 + 3a6z2 - 6a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

{-2, 3}

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 11158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 8 3     j = 1               8 5       j = -1             11 8         j = -3           9 9           j = -5         7 10             j = -7       5 9               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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, 158]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 158]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[2, 10, 3, 9], X[16, 11, 17, 12], X[20, 14, 21, 13], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 158]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -4, 8, -6, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 158]]
Out[5]=	DTCode[4, 10, 18, 14, 2, 16, 20, 8, 22, 6, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 158]][t]
Out[6]=	-4 5 14 25 2 3 4 -29 - t + -- - -- + -- + 25 t - 14 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 158]][z]
Out[7]=	2 4 6 8 1 - 2 z - 4 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 34], Knot[11, Alternating, 158]}
In[9]:=	{KnotDet[Knot[11, Alternating, 158]], KnotSignature[Knot[11, Alternating, 158]]}
Out[9]=	{119, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 158]][q]
Out[10]=	-7 3 7 12 16 19 19 2 3 4 -16 + q - -- + -- - -- + -- - -- + -- + 13 q - 8 q + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 158]}
In[12]:=	A2Invariant[Knot[11, Alternating, 158]][q]
Out[12]=	-20 -18 3 -14 -12 2 5 2 3 -2 2 4 3 + q - q + --- - q - q + --- - -- + -- - -- + q - q + 4 q - 16 10 8 6 4 q q q q q 6 10 12 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 158]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 z 6 - a - 7 a + 3 a + 11 z - ---- - 17 a z + 6 a z + 8 z - -- - 2 2 a a 2 4 4 4 6 2 6 4 6 2 8 > 15 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 158]][a, z]
Out[14]=	2 -2 2 4 z 3 z 3 5 2 6 z 6 + a + 7 a + 3 a - -- - --- - 5 a z - 7 a z - 4 a z - 27 z - ---- - 3 a 2 a a 3 3 2 2 4 2 6 2 8 2 3 z 9 z 3 3 3 > 34 a z - 9 a z + 3 a z - a z + ---- + ---- + 10 a z + 18 a z + 3 a a 4 5 3 7 3 4 15 z 2 4 4 4 6 4 > 12 a z - 2 a z + 56 z + ----- + 65 a z + 17 a z - 6 a z + 2 a 5 5 8 4 3 z 2 z 5 3 5 5 5 7 5 6 > a z - ---- + ---- + 10 a z - 12 a z - 14 a z + 3 a z - 46 z - 3 a a 6 7 7 14 z 2 6 4 6 6 6 z 13 z 7 3 7 > ----- - 57 a z - 19 a z + 6 a z + -- - ----- - 30 a z - 7 a z + 2 3 a a a 8 9 5 7 8 4 z 2 8 4 8 5 z 9 3 9 > 9 a z + 8 z + ---- + 14 a z + 10 a z + ---- + 12 a z + 7 a z + 2 a a 10 2 10 > 2 z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 158]], Vassiliev[3][Knot[11, Alternating, 158]]}
Out[15]=	{-2, 3}
In[16]:=	Kh[Knot[11, Alternating, 158]][q, t]
Out[16]=	9 11 1 2 1 5 2 7 5 9 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 10 9 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 8 q t + 5 q t + 8 q t + 3 q t + 5 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a158

K11a157 K11a159