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

K11a154 K11a156

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 12 2 16 20 8 22 14 6

Alexander Polynomial:

3t-3 - 16t-2 + 40t-1 - 53 + 40t - 16t2 + 3t3

Conway Polynomial:

1 + 3z2 + 2z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{171, -2}

Jones Polynomial:

q-9 - 4q-8 + 9q-7 - 17q-6 + 23q-5 - 27q-4 + 29q-3 - 24q-2 + 19q-1 - 12 + 5q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - 2q-24 + 3q-22 - 5q-20 - 3q-18 + 4q-16 - 4q-14 + 6q-12 + 2q-8 + 4q-6 - 5q-4 + 5q-2 - 3 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 - a2 - 2a2z2 + a2z6 + 6a4 + 10a4z2 + 6a4z4 + 2a4z6 - 5a6 - 6a6z2 - 3a6z4 + a8 + a8z2

Kauffman Polynomial:

a-1z5 - 3z4 + 5z6 - az + 4az3 - 15az5 + 12az7 + a2 - 4a2z2 + 13a2z4 - 25a2z6 + 16a2z8 - 4a3z + 13a3z3 - 18a3z5 - 4a3z7 + 11a3z9 + 6a4 - 16a4z2 + 45a4z4 - 61a4z6 + 24a4z8 + 3a4z10 - 10a5z + 22a5z3 - 7a5z5 - 25a5z7 + 18a5z9 + 5a6 - 15a6z2 + 38a6z4 - 45a6z6 + 15a6z8 + 3a6z10 - 9a7z + 20a7z3 - 14a7z5 - 5a7z7 + 7a7z9 + a8 - 2a8z2 + 7a8z4 - 13a8z6 + 7a8z8 - 2a9z + 7a9z3 - 9a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

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

{3, -4}

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 11155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     4   j = 1                   8 1   j = -1                 11 4     j = -3               14 9       j = -5             15 10         j = -7           12 14           j = -9         11 15             j = -11       6 12               j = -13     3 11                 j = -15   1 6                   j = -17   3                     j = -19 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, 155]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 155]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[12, 7, 13, 8], > X[2, 10, 3, 9], X[16, 11, 17, 12], X[20, 13, 21, 14], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 155]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -8, 5, -2, 6, -4, 7, -10, 8, -6, 9, -3, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 155]]
Out[5]=	DTCode[4, 10, 18, 12, 2, 16, 20, 8, 22, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 155]][t]
Out[6]=	3 16 40 2 3 -53 + -- - -- + -- + 40 t - 16 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 155]][z]
Out[7]=	2 4 6 1 + 3 z + 2 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 155]}
In[9]:=	{KnotDet[Knot[11, Alternating, 155]], KnotSignature[Knot[11, Alternating, 155]]}
Out[9]=	{171, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 155]][q]
Out[10]=	-9 4 9 17 23 27 29 24 19 2 -12 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q 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, 155]}
In[12]:=	A2Invariant[Knot[11, Alternating, 155]][q]
Out[12]=	-28 2 3 5 3 4 4 6 2 4 5 5 -3 + q - --- + --- - --- - --- + --- - --- + --- + -- + -- - -- + -- - 24 22 20 18 16 14 12 8 6 4 2 q q q q q q q q q q q 2 4 6 > 2 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 155]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 4 4 4 -a + 6 a - 5 a + a - 2 a z + 10 a z - 6 a z + a z - z + 6 a z - 6 4 2 6 4 6 > 3 a z + a z + 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 155]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 2 2 a + 6 a + 5 a + a - a z - 4 a z - 10 a z - 9 a z - 2 a z - 4 a z - 4 2 6 2 8 2 10 2 3 3 3 5 3 > 16 a z - 15 a z - 2 a z + a z + 4 a z + 13 a z + 22 a z + 7 3 9 3 4 2 4 4 4 6 4 8 4 > 20 a z + 7 a z - 3 z + 13 a z + 45 a z + 38 a z + 7 a z - 5 10 4 z 5 3 5 5 5 7 5 9 5 6 > 2 a z + -- - 15 a z - 18 a z - 7 a z - 14 a z - 9 a z + 5 z - a 2 6 4 6 6 6 8 6 10 6 7 3 7 > 25 a z - 61 a z - 45 a z - 13 a z + a z + 12 a z - 4 a z - 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 25 a z - 5 a z + 4 a z + 16 a z + 24 a z + 15 a z + 7 a z + 3 9 5 9 7 9 4 10 6 10 > 11 a z + 18 a z + 7 a z + 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 155]], Vassiliev[3][Knot[11, Alternating, 155]]}
Out[15]=	{3, -4}
In[16]:=	Kh[Knot[11, Alternating, 155]][q, t]
Out[16]=	9 11 1 3 1 6 3 11 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 12 11 15 12 14 15 10 14 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a155

K11a154 K11a156