9 41

9_40

9_42

Visit 9 41's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 41's page at Knotilus! Visit 9 41's page at the original Knot Atlas!

9 41 Quick Notes

Three-fold symmetric decorative knot

Three-fold symmetric decorative knot in circle

Knot presentations

Planar diagram presentation	X6271 X12,8,13,7 X14,5,15,6 X10,3,11,4 X2,11,3,12 X4,15,5,16 X8,17,9,18 X16,9,17,10 X18,14,1,13
Gauss code	1, -5, 4, -6, 3, -1, 2, -7, 8, -4, 5, -2, 9, -3, 6, -8, 7, -9
Dowker-Thistlethwaite code	6 10 14 12 16 2 18 4 8
Conway Notation	[20:20:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-7][-4] Hyperbolic Volume 12.0989 A-Polynomial See Data:9 41/A-polynomial

[edit Notes for 9 41's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 41's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-12t+19-12t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{7,t+1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$-q^{3}+3q^{2}-5q+8-8q^{-1}+8q^{-2}-7q^{-3}+5q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-3z^{2}a^{4}-3a^{4}+2z^{4}a^{2}+4z^{2}a^{2}+3a^{2}+z^{4}-z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{8}+2a^{2}z^{8}+3a^{5}z^{7}+9a^{3}z^{7}+6az^{7}+a^{6}z^{6}-a^{4}z^{6}+5a^{2}z^{6}+7z^{6}-10a^{5}z^{5}-26a^{3}z^{5}-11az^{5}+5z^{5}a^{-1}-3a^{6}z^{4}-12a^{4}z^{4}-23a^{2}z^{4}+3z^{4}a^{-2}-11z^{4}+9a^{5}z^{3}+19a^{3}z^{3}+6az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+3a^{6}z^{2}+13a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-2}+6z^{2}-2a^{5}z-4a^{3}z-2az-a^{6}-3a^{4}-3a^{2}$
The A2 invariant	$q^{20}+q^{18}-2q^{16}-q^{12}-2q^{10}+2q^{8}+2q^{4}+q^{2}+2q^{-2}-2q^{-4}+q^{-6}+q^{-8}-q^{-10}$
The G2 invariant	$q^{94}-2q^{92}+6q^{90}-10q^{88}+11q^{86}-7q^{84}-7q^{82}+27q^{80}-39q^{78}+44q^{76}-28q^{74}-3q^{72}+40q^{70}-66q^{68}+70q^{66}-45q^{64}+q^{62}+37q^{60}-64q^{58}+54q^{56}-25q^{54}-16q^{52}+42q^{50}-49q^{48}+27q^{46}+7q^{44}-43q^{42}+63q^{40}-60q^{38}+37q^{36}+6q^{34}-46q^{32}+79q^{30}-82q^{28}+64q^{26}-19q^{24}-28q^{22}+65q^{20}-77q^{18}+58q^{16}-17q^{14}-25q^{12}+51q^{10}-48q^{8}+18q^{6}+19q^{4}-46q^{2}+51-30q^{-2}-3q^{-4}+35q^{-6}-51q^{-8}+53q^{-10}-32q^{-12}+8q^{-14}+16q^{-16}-32q^{-18}+33q^{-20}-27q^{-22}+18q^{-24}-7q^{-26}-2q^{-28}+9q^{-30}-14q^{-32}+13q^{-34}-10q^{-36}+7q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+3q^{-48}-2q^{-50}+q^{-52}$

Further Quantum Invariants

Further quantum knot invariants for 9_41.

A1 Invariants.

Weight	Invariant
1	$q^{13}-2q^{11}+2q^{9}-2q^{7}+q^{5}+3q^{-1}-2q^{-3}+2q^{-5}-q^{-7}$
2	$q^{38}-2q^{36}-3q^{34}+7q^{32}+q^{30}-11q^{28}+7q^{26}+9q^{24}-13q^{22}+12q^{18}-8q^{16}-6q^{14}+9q^{12}-8q^{8}+2q^{6}+9q^{4}-5q^{2}-8+14q^{-2}+q^{-4}-13q^{-6}+9q^{-8}+4q^{-10}-7q^{-12}+3q^{-14}-2q^{-18}+q^{-20}$
3	$q^{75}-2q^{73}-3q^{71}+2q^{69}+10q^{67}+4q^{65}-18q^{63}-16q^{61}+16q^{59}+34q^{57}-4q^{55}-47q^{53}-17q^{51}+46q^{49}+41q^{47}-34q^{45}-60q^{43}+15q^{41}+66q^{39}+9q^{37}-62q^{35}-28q^{33}+55q^{31}+39q^{29}-43q^{27}-46q^{25}+32q^{23}+50q^{21}-19q^{19}-53q^{17}+7q^{15}+49q^{13}+11q^{11}-47q^{9}-33q^{7}+32q^{5}+57q^{3}-11q-66q^{-1}-11q^{-3}+66q^{-5}+35q^{-7}-56q^{-9}-42q^{-11}+34q^{-13}+40q^{-15}-15q^{-17}-26q^{-19}+5q^{-21}+14q^{-23}-4q^{-25}-4q^{-27}+q^{-31}-q^{-33}+2q^{-37}-q^{-39}$
4	$q^{124}-2q^{122}-3q^{120}+2q^{118}+5q^{116}+13q^{114}-3q^{112}-23q^{110}-24q^{108}-8q^{106}+55q^{104}+57q^{102}+3q^{100}-72q^{98}-121q^{96}-2q^{94}+116q^{92}+159q^{90}+43q^{88}-191q^{86}-202q^{84}-47q^{82}+217q^{80}+294q^{78}+9q^{76}-254q^{74}-328q^{72}-12q^{70}+352q^{68}+302q^{66}-22q^{64}-398q^{62}-304q^{60}+142q^{58}+393q^{56}+244q^{54}-237q^{52}-398q^{50}-86q^{48}+293q^{46}+335q^{44}-67q^{42}-340q^{40}-178q^{38}+193q^{36}+310q^{34}+9q^{32}-275q^{30}-205q^{28}+130q^{26}+287q^{24}+96q^{22}-210q^{20}-278q^{18}-8q^{16}+244q^{14}+274q^{12}-17q^{10}-326q^{8}-273q^{6}+42q^{4}+409q^{2}+310-163q^{-2}-441q^{-4}-287q^{-6}+269q^{-8}+479q^{-10}+133q^{-12}-301q^{-14}-416q^{-16}-3q^{-18}+316q^{-20}+233q^{-22}-46q^{-24}-251q^{-26}-98q^{-28}+88q^{-30}+116q^{-32}+38q^{-34}-76q^{-36}-41q^{-38}+9q^{-40}+20q^{-42}+20q^{-44}-17q^{-46}-6q^{-48}+2q^{-50}+6q^{-54}-3q^{-56}+q^{-58}-2q^{-62}+q^{-64}$
5	$q^{185}-2q^{183}-3q^{181}+2q^{179}+5q^{177}+8q^{175}+6q^{173}-8q^{171}-31q^{169}-30q^{167}-2q^{165}+44q^{163}+84q^{161}+72q^{159}-17q^{157}-143q^{155}-190q^{153}-106q^{151}+98q^{149}+309q^{147}+346q^{145}+105q^{143}-294q^{141}-567q^{139}-486q^{137}+2q^{135}+617q^{133}+899q^{131}+537q^{129}-314q^{127}-1081q^{125}-1177q^{123}-371q^{121}+857q^{119}+1618q^{117}+1232q^{115}-157q^{113}-1597q^{111}-1983q^{109}-858q^{107}+1052q^{105}+2327q^{103}+1871q^{101}-92q^{99}-2114q^{97}-2599q^{95}-1026q^{93}+1445q^{91}+2851q^{89}+1988q^{87}-505q^{85}-2620q^{83}-2615q^{81}-445q^{79}+2075q^{77}+2828q^{75}+1183q^{73}-1396q^{71}-2689q^{69}-1637q^{67}+776q^{65}+2358q^{63}+1770q^{61}-329q^{59}-1961q^{57}-1691q^{55}+88q^{53}+1632q^{51}+1513q^{49}-23q^{47}-1421q^{45}-1360q^{43}+34q^{41}+1345q^{39}+1316q^{37}+17q^{35}-1317q^{33}-1448q^{31}-230q^{29}+1238q^{27}+1676q^{25}+701q^{23}-931q^{21}-1927q^{19}-1397q^{17}+334q^{15}+1957q^{13}+2167q^{11}+614q^{9}-1637q^{7}-2789q^{5}-1726q^{3}+888q+2982q^{-1}+2752q^{-3}+198q^{-5}-2647q^{-7}-3387q^{-9}-1306q^{-11}+1844q^{-13}+3402q^{-15}+2149q^{-17}-804q^{-19}-2892q^{-21}-2461q^{-23}-109q^{-25}+2034q^{-27}+2259q^{-29}+678q^{-31}-1154q^{-33}-1723q^{-35}-843q^{-37}+492q^{-39}+1118q^{-41}+702q^{-43}-119q^{-45}-603q^{-47}-478q^{-49}-30q^{-51}+297q^{-53}+265q^{-55}+36q^{-57}-118q^{-59}-124q^{-61}-33q^{-63}+51q^{-65}+55q^{-67}+9q^{-69}-23q^{-71}-17q^{-73}+q^{-75}+6q^{-77}+8q^{-79}+2q^{-81}-6q^{-83}-4q^{-85}+3q^{-87}-q^{-89}+2q^{-93}-q^{-95}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-2q^{16}-q^{12}-2q^{10}+2q^{8}+2q^{4}+q^{2}+2q^{-2}-2q^{-4}+q^{-6}+q^{-8}-q^{-10}$
1,1	$q^{52}-4q^{50}+14q^{48}-36q^{46}+66q^{44}-110q^{42}+156q^{40}-196q^{38}+224q^{36}-216q^{34}+182q^{32}-112q^{30}+22q^{28}+80q^{26}-188q^{24}+272q^{22}-339q^{20}+376q^{18}-378q^{16}+346q^{14}-280q^{12}+190q^{10}-96q^{8}-4q^{6}+84q^{4}-134q^{2}+168-158q^{-2}+149q^{-4}-122q^{-6}+98q^{-8}-78q^{-10}+56q^{-12}-46q^{-14}+38q^{-16}-28q^{-18}+18q^{-20}-12q^{-22}+8q^{-24}-4q^{-26}+q^{-28}$
2,0	$q^{52}+q^{50}-q^{48}-5q^{46}-3q^{44}+4q^{42}+5q^{40}-2q^{38}-3q^{36}+6q^{34}+8q^{32}-3q^{30}-8q^{28}+2q^{26}+2q^{24}-4q^{22}-7q^{20}+3q^{18}+4q^{16}-3q^{14}+q^{12}-q^{8}+q^{6}+6q^{4}-3q^{2}-3+7q^{-2}+8q^{-4}-5q^{-6}-8q^{-8}+7q^{-10}+5q^{-12}-5q^{-14}-3q^{-16}+2q^{-18}+2q^{-20}-2q^{-22}-q^{-24}+q^{-26}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}-2q^{38}+2q^{36}+2q^{34}-6q^{32}+6q^{30}-8q^{26}+8q^{24}-q^{22}-8q^{20}+7q^{18}+2q^{16}-7q^{14}+2q^{12}+2q^{10}-q^{8}-3q^{6}+2q^{4}+9q^{2}-5+q^{-2}+11q^{-4}-9q^{-6}-2q^{-8}+8q^{-10}-5q^{-12}-3q^{-14}+5q^{-16}-q^{-18}-2q^{-20}+q^{-22}$
1,0,0	$q^{27}+q^{25}+q^{23}-2q^{21}-3q^{17}-q^{15}-2q^{13}+2q^{11}+q^{9}+2q^{7}+2q^{5}+q^{3}+q-q^{-1}+2q^{-3}-2q^{-5}+q^{-7}+q^{-11}-q^{-13}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}-2q^{38}+6q^{36}-8q^{34}+10q^{32}-12q^{30}+12q^{28}-12q^{26}+8q^{24}-5q^{22}-2q^{20}+7q^{18}-14q^{16}+19q^{14}-22q^{12}+24q^{10}-21q^{8}+19q^{6}-12q^{4}+9q^{2}-1-3q^{-2}+9q^{-4}-11q^{-6}+12q^{-8}-12q^{-10}+9q^{-12}-7q^{-14}+5q^{-16}-3q^{-18}+2q^{-20}-q^{-22}$
1,0	$q^{66}-2q^{62}-2q^{60}+4q^{58}+5q^{56}-4q^{54}-8q^{52}+11q^{48}+5q^{46}-11q^{44}-10q^{42}+7q^{40}+13q^{38}-13q^{34}-5q^{32}+9q^{30}+8q^{28}-5q^{26}-8q^{24}+2q^{22}+7q^{20}-q^{18}-9q^{16}-q^{14}+9q^{12}+3q^{10}-9q^{8}-5q^{6}+8q^{4}+11q^{2}-4-10q^{-2}+2q^{-4}+14q^{-6}+3q^{-8}-10q^{-10}-8q^{-12}+5q^{-14}+10q^{-16}-7q^{-20}-5q^{-22}+2q^{-24}+5q^{-26}+q^{-28}-2q^{-30}-2q^{-32}+q^{-36}$

G2 Invariants.

Weight

Invariant

1,0

$q^{94}-2q^{92}+6q^{90}-10q^{88}+11q^{86}-7q^{84}-7q^{82}+27q^{80}-39q^{78}+44q^{76}-28q^{74}-3q^{72}+40q^{70}-66q^{68}+70q^{66}-45q^{64}+q^{62}+37q^{60}-64q^{58}+54q^{56}-25q^{54}-16q^{52}+42q^{50}-49q^{48}+27q^{46}+7q^{44}-43q^{42}+63q^{40}-60q^{38}+37q^{36}+6q^{34}-46q^{32}+79q^{30}-82q^{28}+64q^{26}-19q^{24}-28q^{22}+65q^{20}-77q^{18}+58q^{16}-17q^{14}-25q^{12}+51q^{10}-48q^{8}+18q^{6}+19q^{4}-46q^{2}+51-30q^{-2}-3q^{-4}+35q^{-6}-51q^{-8}+53q^{-10}-32q^{-12}+8q^{-14}+16q^{-16}-32q^{-18}+33q^{-20}-27q^{-22}+18q^{-24}-7q^{-26}-2q^{-28}+9q^{-30}-14q^{-32}+13q^{-34}-10q^{-36}+7q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+3q^{-48}-2q^{-50}+q^{-52}$

.

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["9 41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-12t+19-12t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+1$

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]=

$\{7,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{3}+3q^{2}-5q+8-8q^{-1}+8q^{-2}-7q^{-3}+5q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-3z^{2}a^{4}-3a^{4}+2z^{4}a^{2}+4z^{2}a^{2}+3a^{2}+z^{4}-z^{2}a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{8}+2a^{2}z^{8}+3a^{5}z^{7}+9a^{3}z^{7}+6az^{7}+a^{6}z^{6}-a^{4}z^{6}+5a^{2}z^{6}+7z^{6}-10a^{5}z^{5}-26a^{3}z^{5}-11az^{5}+5z^{5}a^{-1}-3a^{6}z^{4}-12a^{4}z^{4}-23a^{2}z^{4}+3z^{4}a^{-2}-11z^{4}+9a^{5}z^{3}+19a^{3}z^{3}+6az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+3a^{6}z^{2}+13a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-2}+6z^{2}-2a^{5}z-4a^{3}z-2az-a^{6}-3a^{4}-3a^{2}$

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 $0$ $8$ $0$ $-48$ $-24$ $0$ ${\frac {368}{3}}$ $-{\frac {64}{3}}$ $104$ $0$ $32$ $0$ $0$ $-136$ $296$ $-328$ $-88$ $-72$

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=$0 is the signature of 9 41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	3	χ
7										1	-1
5									2		2
3								3	1		-2
1							5	2			3
-1						4	4				0
-3					4	4					0
-5				3	4						1
-7			2	4							-2
-9		1	3								2
-11		2									-2
-13	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[9, 41]]
Out[2]=	9
In[3]:=	PD[Knot[9, 41]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[14, 5, 15, 6], X[10, 3, 11, 4], X[2, 11, 3, 12], X[4, 15, 5, 16], X[8, 17, 9, 18], X[16, 9, 17, 10], X[18, 14, 1, 13]]
In[4]:=	GaussCode[Knot[9, 41]]
Out[4]=	GaussCode[1, -5, 4, -6, 3, -1, 2, -7, 8, -4, 5, -2, 9, -3, 6, -8, 7, -9]
In[5]:=	BR[Knot[9, 41]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	alex = Alexander[Knot[9, 41]][t]
Out[6]=	3 12 2 19 + -- - -- - 12 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 41]][z]
Out[7]=	4 1 + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 41], Knot[11, NonAlternating, 83]}
In[9]:=	{KnotDet[Knot[9, 41]], KnotSignature[Knot[9, 41]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[9, 41]][q]
Out[10]=	-6 3 5 7 8 8 2 3 8 + q - -- + -- - -- + -- - - - 5 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 41], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21]}
In[12]:=	A2Invariant[Knot[9, 41]][q]
Out[12]=	-20 -18 2 -12 2 2 2 -2 2 4 6 q + q - --- - q - --- + -- + -- + q + 2 q - 2 q + q + 16 10 8 4 q q q q 8 10 q - q
In[13]:=	Kauffman[Knot[9, 41]][a, z]
Out[13]=	2 2 4 6 3 5 2 z 2 2 -3 a - 3 a - a - 2 a z - 4 a z - 2 a z + 6 z - -- + 17 a z + 2 a 3 3 4 2 6 2 z 3 z 3 3 3 5 3 13 a z + 3 a z + -- - ---- + 6 a z + 19 a z + 9 a z - 3 a a 4 5 4 3 z 2 4 4 4 6 4 5 z 5 11 z + ---- - 23 a z - 12 a z - 3 a z + ---- - 11 a z - 2 a a 3 5 5 5 6 2 6 4 6 6 6 7 26 a z - 10 a z + 7 z + 5 a z - a z + a z + 6 a z + 3 7 5 7 2 8 4 8 9 a z + 3 a z + 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[9, 41]], Vassiliev[3][Knot[9, 41]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[9, 41]][q, t]
Out[15]=	4 1 2 1 3 2 4 3 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 4 4 4 4 3 3 2 5 2 7 3 ----- + ----- + ---- + --- + 2 q t + 3 q t + q t + 2 q t + q t 5 2 3 2 3 q t q t q t q t