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

K11a150 K11a152

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 20 14 2 18 22 12 6 8

Alexander Polynomial:

- t-4 + 6t-3 - 15t-2 + 26t-1 - 31 + 26t - 15t2 + 6t3 - t4

Conway Polynomial:

1 + 4z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{127, -2}

Jones Polynomial:

- q-8 + 3q-7 - 8q-6 + 13q-5 - 17q-4 + 21q-3 - 20q-2 + 18q-1 - 13 + 8q - 4q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-20 - 3q-18 + 3q-16 - 2q-14 + 3q-12 + 3q-10 - q-8 + 5q-6 - 4q-4 + 3q-2 - 1 - 2q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2z2 + 3z4 + z6 - a2 - 6a2z2 - 9a2z4 - 5a2z6 - a2z8 + 5a4 + 11a4z2 + 8a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 5a-1z3 - 10a-1z5 + 4a-1z7 - 4z2 + 15z4 - 19z6 + 7z8 - az + 3az3 + 2az5 - 11az7 + 6az9 + a2 - 10a2z2 + 30a2z4 - 32a2z6 + 9a2z8 + 2a2z10 - 4a3z + 5a3z3 + 10a3z5 - 21a3z7 + 11a3z9 + 5a4 - 14a4z2 + 25a4z4 - 25a4z6 + 9a4z8 + 2a4z10 - 8a5z + 17a5z3 - 13a5z5 + 5a5z9 + 3a6 - 7a6z2 + 8a6z4 - 10a6z6 + 7a6z8 - 4a7z + 8a7z3 - 10a7z5 + 6a7z7 + a8z2 - 4a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{4, -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 11151. 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             11 9         j = -5           10 9           j = -7         7 11             j = -9       6 10               j = -11     2 7                 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, 151]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 151]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[20, 7, 21, 8], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[18, 14, 19, 13], X[22, 15, 1, 16], > X[12, 18, 13, 17], X[6, 19, 7, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 151]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -5, 8, -3, 9, -7, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 151]]
Out[5]=	DTCode[4, 10, 16, 20, 14, 2, 18, 22, 12, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 151]][t]
Out[6]=	-4 6 15 26 2 3 4 -31 - t + -- - -- + -- + 26 t - 15 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 151]][z]
Out[7]=	2 4 6 8 1 + 4 z + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 151]}
In[9]:=	{KnotDet[Knot[11, Alternating, 151]], KnotSignature[Knot[11, Alternating, 151]]}
Out[9]=	{127, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 151]][q]
Out[10]=	-8 3 8 13 17 21 20 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, 151]}
In[12]:=	A2Invariant[Knot[11, Alternating, 151]][q]
Out[12]=	-24 -20 3 3 2 3 3 -8 5 4 3 2 -1 - q - q - --- + --- - --- + --- + --- - q + -- - -- + -- - 2 q + 18 16 14 12 10 6 4 2 q q q q q q q q 4 6 8 > 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 151]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 -a + 5 a - 3 a + 2 z - 6 a z + 11 a z - 3 a z + 3 z - 9 a z + 4 4 6 4 6 2 6 4 6 2 8 > 8 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 151]][a, z]
Out[14]=	2 4 6 3 5 7 9 2 2 2 a + 5 a + 3 a - a z - 4 a z - 8 a z - 4 a z + a z - 4 z - 10 a z - 3 4 2 6 2 8 2 5 z 3 3 3 5 3 7 3 > 14 a z - 7 a z + a z + ---- + 3 a z + 5 a z + 17 a z + 8 a z - a 4 5 9 3 4 2 z 2 4 4 4 6 4 8 4 10 z > 2 a z + 15 z - ---- + 30 a z + 25 a z + 8 a z - 4 a z - ----- + 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 2 a z + 10 a z - 13 a z - 10 a z + a z - 19 z + -- - 32 a z - 2 a 7 4 6 6 6 8 6 4 z 7 3 7 7 7 > 25 a z - 10 a z + 3 a z + ---- - 11 a z - 21 a z + 6 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 7 z + 9 a z + 9 a z + 7 a z + 6 a z + 11 a z + 5 a z + 2 10 4 10 > 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 151]], Vassiliev[3][Knot[11, Alternating, 151]]}
Out[15]=	{4, -7}
In[16]:=	Kh[Knot[11, Alternating, 151]][q, t]
Out[16]=	9 10 1 2 1 6 2 7 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 7 11 10 9 11 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 K11a151

K11a150 K11a152