10 136

10_135

10_137

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10 136 Quick Notes

10 136 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code	-1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 -14 2 -18 -6 -20 -12 -16
Conway Notation	[22,22,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-6] Hyperbolic Volume 7.74627 A-Polynomial See Data:10 136/A-polynomial

[edit Notes for 10 136's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 136's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+4t-5+4t^{-1}-t^{-2}$
Conway polynomial	$1-z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, 2 }
Jones polynomial	$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}+a^{2}+3a^{-2}-a^{-4}-2$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+3z^{7}a^{-1}+z^{7}a^{-3}+a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-9az^{5}-14z^{5}a^{-1}-5z^{5}a^{-3}-4a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}+9az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}+3a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-2az-4za^{-1}-2za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$
The A2 invariant	$q^{10}-q^{2}+q^{-4}+2q^{-6}+q^{-8}+q^{-10}-q^{-12}-q^{-14}$
The G2 invariant	$q^{46}-q^{44}+2q^{42}-2q^{40}+q^{38}-2q^{34}+6q^{32}-4q^{30}+3q^{28}-2q^{24}+3q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+3q^{10}-6q^{8}+6q^{6}-7q^{4}+3-5q^{-2}+4q^{-4}-3q^{-6}+3q^{-8}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}+q^{-24}+3q^{-26}+4q^{-30}-6q^{-32}+6q^{-34}-2q^{-36}+4q^{-40}-7q^{-42}+6q^{-44}-2q^{-48}+q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-3q^{-58}+q^{-60}+q^{-62}-3q^{-64}+3q^{-66}-3q^{-68}+q^{-70}+q^{-72}-2q^{-74}+q^{-76}$

Further Quantum Invariants

Further quantum knot invariants for 10_136.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{-1}+q^{-3}+q^{-7}-q^{-9}$
2	$q^{22}-q^{20}-2q^{18}+2q^{16}+q^{14}-q^{12}+q^{8}+q^{6}-2q^{4}+2-q^{-2}+2q^{-6}-q^{-8}+2q^{-12}+q^{-14}-q^{-16}-q^{-18}+q^{-20}-2q^{-24}+q^{-28}$
3	$q^{45}-q^{43}-2q^{41}+3q^{37}+3q^{35}-3q^{33}-3q^{31}+2q^{27}+2q^{25}-2q^{21}-4q^{19}+q^{17}+6q^{15}+2q^{13}-6q^{11}-4q^{9}+5q^{7}+6q^{5}-3q^{3}-5q+2q^{-1}+6q^{-3}-q^{-5}-4q^{-7}+q^{-9}+3q^{-11}-q^{-13}-2q^{-15}+q^{-17}+q^{-19}+2q^{-21}-4q^{-25}-q^{-27}+7q^{-29}+5q^{-31}-7q^{-33}-10q^{-35}+5q^{-37}+8q^{-39}-2q^{-41}-7q^{-43}-q^{-45}+5q^{-47}+2q^{-49}-q^{-51}-q^{-53}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}-q^{2}+q^{-4}+2q^{-6}+q^{-8}+q^{-10}-q^{-12}-q^{-14}$
2,0	$q^{28}-q^{24}-q^{22}+q^{18}+2q^{8}+q^{6}+q^{2}+1-2q^{-4}-q^{-6}-q^{-8}-q^{-10}+2q^{-14}+4q^{-16}+3q^{-18}+4q^{-20}-q^{-22}-q^{-24}-2q^{-26}-2q^{-28}-2q^{-30}-2q^{-32}+q^{-34}+q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{14}+q^{10}+q^{6}-q^{4}-q^{2}-q^{-2}+q^{-6}+3q^{-8}+q^{-10}+3q^{-12}+q^{-14}-q^{-18}-q^{-20}-q^{-24}$
1,0,0	$q^{13}+q^{9}-q^{3}-q-q^{-1}+q^{-5}+2q^{-7}+3q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-17}-q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+2q^{16}-q^{14}+2q^{12}-q^{10}-q^{6}-q^{4}+q^{2}-4+3q^{-2}-2q^{-4}+3q^{-6}-q^{-8}+3q^{-10}+q^{-12}+q^{-14}+2q^{-16}-q^{-18}+q^{-20}-2q^{-22}+q^{-24}-2q^{-26}$
1,0	$q^{34}-q^{30}-q^{28}+q^{26}+q^{24}-q^{22}-q^{20}+2q^{18}+2q^{16}-q^{14}-q^{12}+2q^{8}-2q^{4}-q^{2}+1-q^{-4}+2q^{-10}+q^{-12}+3q^{-18}+2q^{-20}-2q^{-24}+q^{-26}+q^{-28}-q^{-30}-2q^{-32}+q^{-36}-q^{-40}-q^{-42}+q^{-44}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+2q^{42}-2q^{40}+q^{38}-2q^{34}+6q^{32}-4q^{30}+3q^{28}-2q^{24}+3q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+3q^{10}-6q^{8}+6q^{6}-7q^{4}+3-5q^{-2}+4q^{-4}-3q^{-6}+3q^{-8}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}+q^{-24}+3q^{-26}+4q^{-30}-6q^{-32}+6q^{-34}-2q^{-36}+4q^{-40}-7q^{-42}+6q^{-44}-2q^{-48}+q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-3q^{-58}+q^{-60}+q^{-62}-3q^{-64}+3q^{-66}-3q^{-68}+q^{-70}+q^{-72}-2q^{-74}+q^{-76}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 136"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{2}+4t-5+4t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{4}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}+a^{2}+3a^{-2}-a^{-4}-2$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+3z^{7}a^{-1}+z^{7}a^{-3}+a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-9az^{5}-14z^{5}a^{-1}-5z^{5}a^{-3}-4a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}+9az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}+3a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-2az-4za^{-1}-2za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$

Vassiliev invariants

V2 and V3:

(0, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $8$ $0$ $16$ $8$ $0$ ${\frac {80}{3}}$ $-{\frac {64}{3}}$ $40$ $0$ $32$ $0$ $0$ $88$ $-88$ ${\frac {328}{3}}$ ${\frac {88}{3}}$ $24$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-4	-3	-2	-1	0	1	2	3	χ
9								1	-1
7							1		1
5						1	1		0
3					2	1			1
1				1	2				1
-1			1	2	1				0
-3		1	1						0
-5		1							-1
-7	1								1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 136]]
Out[2]=	10
In[3]:=	PD[Knot[10, 136]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16], X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[10, 136]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7]
In[5]:=	BR[Knot[10, 136]]
Out[5]=	BR[5, {1, -2, 1, -2, -3, 2, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 136]][t]
Out[6]=	-2 4 2 -5 - t + - + 4 t - t t
In[7]:=	Conway[Knot[10, 136]][z]
Out[7]=	4 1 - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 21], Knot[10, 136]}
In[9]:=	{KnotDet[Knot[10, 136]], KnotSignature[Knot[10, 136]]}
Out[9]=	{15, 2}
In[10]:=	J=Jones[Knot[10, 136]][q]
Out[10]=	-3 2 2 2 3 4 -2 + q - -- + - + 3 q - 2 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 136], Knot[11, NonAlternating, 92]}
In[12]:=	A2Invariant[Knot[10, 136]][q]
Out[12]=	-10 -2 4 6 8 10 12 14 q - q + q + 2 q + q + q - q - q
In[13]:=	Kauffman[Knot[10, 136]][a, z]
Out[13]=	2 2 -4 3 2 2 z 4 z 2 z 4 z 2 2 -2 - a - -- - a - --- - --- - 2 a z + 6 z + -- + ---- + 3 a z + 2 3 a 4 2 a a a a 3 3 4 5 5 7 z 16 z 3 4 2 z 2 4 5 z 14 z ---- + ----- + 9 a z - 2 z + ---- - 4 a z - ---- - ----- - 3 a 2 3 a a a a 6 7 7 8 5 6 4 z 2 6 z 3 z 7 8 z 9 a z - 3 z - ---- + a z + -- + ---- + 2 a z + z + -- 2 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 136]], Vassiliev[3][Knot[10, 136]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[10, 136]][q, t]
Out[15]=	1 3 1 1 1 1 1 2 q - + 2 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + - + q 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 9 3 q t + q t + q t + q t + q t