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

K11n133 K11n135

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,19,6,18 X7,22,8,1 X9,14,10,15 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,7,18,6 X19,12,20,13 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 12t-1 - 17 + 12t - 3t2

Conway Polynomial:

1 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{925, ...}

Determinant and Signature:

{47, -2}

Jones Polynomial:

q-9 - 3q-8 + 5q-7 - 7q-6 + 8q-5 - 8q-4 + 7q-3 - 5q-2 + 3q-1

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - q-24 + 2q-22 - q-20 + q-18 - 2q-14 - 2q-10 + 2q-8 + q-6 + 3q-2

HOMFLY-PT Polynomial:

3a2 + 3a2z2 - 3a4 - 4a4z2 - 2a4z4 + a6 - a6z4 + a8z2

Kauffman Polynomial:

- 3a2 + 5a2z2 + a3z - a3z3 + 3a3z5 - 3a4 + 7a4z2 - 2a4z4 + a4z8 + a5z - 3a5z3 + 2a5z5 - a5z7 + a5z9 - a6 + 4a6z4 - 9a6z6 + 4a6z8 - a7z + 6a7z3 - 11a7z5 + 2a7z7 + a7z9 + 3a8z4 - 8a8z6 + 3a8z8 - a9z + 8a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{0, 1}

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 11134. 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 j = -1                 3 j = -3               3 1 j = -5             4 2   j = -7           4 3     j = -9         4 4       j = -11       3 4         j = -13     2 4           j = -15   1 3             j = -17   2               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, NonAlternating, 134]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 134]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 19, 6, 18], X[7, 22, 8, 1], > X[9, 14, 10, 15], X[2, 11, 3, 12], X[13, 20, 14, 21], X[15, 8, 16, 9], > X[17, 7, 18, 6], X[19, 12, 20, 13], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 134]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 9, -4, 8, -5, -2, 6, 10, -7, 5, -8, 11, -9, 3, -10, > 7, -11, 4]
In[5]:=	DTCode[Knot[11, NonAlternating, 134]]
Out[5]=	DTCode[4, 10, -18, -22, -14, 2, -20, -8, -6, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 134]][t]
Out[6]=	3 12 2 -17 - -- + -- + 12 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 134]][z]
Out[7]=	4 1 - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 25], Knot[11, NonAlternating, 134]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 134]], KnotSignature[Knot[11, NonAlternating, 134]]}
Out[9]=	{47, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 134]][q]
Out[10]=	-9 3 5 7 8 8 7 5 3 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, NonAlternating, 134]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 134]][q]
Out[12]=	-28 -26 -24 2 -20 -18 2 2 2 -6 3 q - q - q + --- - q + q - --- - --- + -- + q + -- 22 14 10 8 2 q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 134]][a, z]
Out[13]=	2 4 6 2 2 4 2 8 2 4 4 6 4 3 a - 3 a + a + 3 a z - 4 a z + a z - 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 134]][a, z]
Out[14]=	2 4 6 3 5 7 9 2 2 4 2 10 2 -3 a - 3 a - a + a z + a z - a z - a z + 5 a z + 7 a z + 2 a z - 3 3 5 3 7 3 9 3 4 4 6 4 8 4 > a z - 3 a z + 6 a z + 8 a z - 2 a z + 4 a z + 3 a z - 10 4 3 5 5 5 7 5 9 5 6 6 8 6 > 3 a z + 3 a z + 2 a z - 11 a z - 10 a z - 9 a z - 8 a z + 10 6 5 7 7 7 9 7 4 8 6 8 8 8 5 9 > a z - a z + 2 a z + 3 a z + a z + 4 a z + 3 a z + a z + 7 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 134]], Vassiliev[3][Knot[11, NonAlternating, 134]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 134]][q, t]
Out[16]=	-3 3 1 2 1 3 2 4 3 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 4 4 4 4 3 4 2 3 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n134

K11n133 K11n135